Ordinary Differential Equations and Sturm–Liouville Theory

Derivation. Derives Lemma 9.6. Fix \(x \in S\). Each component obeys \(\abs{u_{j}^{i}(x)} \le \norm{\vect{u}_{j}(x)} \le M_{j}\), so \(\sum_{j}\abs{u_{j}^{i}(x)}\) converges by comparison with \(\sum_{j}M_{j}\) and the component series converges, both by Proposition 7.47. Call the limit \(\vect{U}(x)\).

For the uniformity, subtract the \(k\)-th partial sum: for \(k \ge N\),

and the right-hand side is the tail of a convergent series of numbers, hence below any prescribed \(\varepsilon\) once \(N\) is large enough. It contains no \(x\), which is exactly Equation (9.5).

Suppose now every \(\vect{u}_{j}\) is continuous and fix \(x_{0}\in S\) and \(\varepsilon>0\). Choose \(N\) with \(\sum_{j>N}M_{j} < \varepsilon/3\) and write \(\vect{S}_{N} = \sum_{j=0}^{N}\vect{u}_{j}\), a finite sum of continuous maps and therefore continuous (Definition 7.20): there is a \(\delta>0\) with \(\norm{\vect{S}_{N}(x)-\vect{S}_{N}(x_{0})} < \varepsilon/3\) for every \(x \in S\) with \(\abs{x-x_{0}}<\delta\). For such an \(x\),

the outer two terms being bounded by \(\sum_{j>N}M_{j}\) by the estimate just proved. Hence \(\vect{U}\) is continuous at \(x_{0}\).

Derivation. Derives Theorem 9.8. A continuous \(\vect{Y}\) solves Equation (9.6) if and only if it solves the integral equation

by the fundamental theorem of calculus (Theorems 7.42 and 7.43): differentiating Equation (9.7) returns Equation (9.6) and evaluating it at \(x_{0}\) returns the initial value, while integrating Equation (9.6) from \(x_{0}\) to \(x\) returns Equation (9.7).

Define the Picard iterates by \(\vect{Y}_{0}(x) = \vect{Y}_{0}\) and

Every iterate stays inside the box on \(\abs{x-x_{0}}\le h\): if \(\norm{\vect{Y}_{k}(s) - \vect{Y}_{0}} \le b\) there, the integrand of Equation (9.8) is defined and bounded by \(M\), so \(\norm{\vect{Y}_{k+1}(x) - \vect{Y}_{0}} \le M\abs{x-x_{0}} \le Mh \le b\); and the hypothesis holds trivially for \(k = 0\).

The successive differences obey

For \(k = 0\) this is the bound just used. Assuming it at \(k-1\) and applying Equation (9.4),

which is Equation (9.9).

Write \(\vect{Y}_{k} = \vect{Y}_{0} + \sum_{j=0}^{k-1}\bigl(\vect{Y}_{j+1} - \vect{Y}_{j}\bigr)\). By Equation (9.9) the terms of that series are bounded in supremum norm on \(\abs{x-x_{0}} \le h\) by \(M L^{j}h^{j+1}/(j+1)!\), whose sum is \(M\bigl(\ee^{Lh}-1\bigr)/L\), a finite number; these are the constants \(M_{j}\) of Lemma 9.6, so the series converges absolutely and uniformly on \(\abs{x-x_{0}}\le h\). Its sum \(\vect{Y}\) is therefore continuous, by the last clause of the same lemma, and obeys \(\norm{\vect{Y}-\vect{Y}_{0}} \le b\). Passing to the limit under the integral sign in Equation (9.8) is legitimate because Equation (9.4) gives \(\norm{\vect{f}(s,\vect{Y}_{k}) - \vect{f}(s,\vect{Y})} \le L\,\norm{\vect{Y}_{k}-\vect{Y}}\), which tends to zero uniformly by Equation (9.5), while the error committed on the integral is at most \(h\) times that supremum. Hence \(\vect{Y}\) solves Equation (9.7).

For uniqueness let \(\vect{Y}\) and \(\tilde{\vect{Y}}\) both solve Equation (9.7) on \(\abs{x-x_{0}} \le h\) and put \(\delta = \sup\norm{\vect{Y}-\tilde{\vect{Y}}}\), finite because both stay in the box. Subtracting the two integral equations and using Equation (9.4) gives \(\norm{\vect{Y}(x)-\tilde{\vect{Y}}(x)} \le L\,\delta\,\abs{x-x_{0}}\); feeding that bound back into the same inequality gives \(L^{2}\delta\abs{x-x_{0}}^{2}/2\), and after \(k\) rounds

which tends to zero as \(k \to \infty\) because the exponential series Equation (7.31) converges. Hence \(\delta = 0\).

Derivation. Derives Corollary 9.9. Here \(\vect{f}(x,\vect{Y}) = A(x)\vect{Y}+\vect{g}(x)\) is defined for every \(\vect{Y}\), so the box of Definition 9.4 may be taken to be \(J\times\R^{n}\) and Equation (9.4) holds globally with \(L = \max_{J}\norm{A}\), the maximum existing because \(J\) is compact and \(A\) continuous (Theorem 7.24). The construction in the proof of Theorem 9.8 then runs on all of \(J\) without the restriction \(h \le b/M\), which was needed only to keep the iterates inside a bounded box: the estimate Equation (9.9) holds on \(J\) with \(M = \max_{J}\norm{\vect{f}(\cdot,\vect{Y}_{0})}\), and the comparison series is again the exponential one. Uniqueness is unchanged. The scalar statement follows through Equation (9.3), whose coefficients \(-a_{i}/a_{n}\) and inhomogeneity \(b/a_{n}\) are continuous precisely because \(a_{n}\) does not vanish.

Derivation. Derives Lemma 9.10. Write \(U(x) = \alpha + \beta\int_{x_{0}}^{x}u(s)\,\dd s\). Since \(u\) is continuous, \(U\) is differentiable with \(U' = \beta u\) (Theorem 7.42), and \(U(x_{0}) = \alpha\). The hypothesis Equation (9.10) is exactly \(u \le U\), so \(U' = \beta u \le \beta U\) and

A function whose derivative is nowhere positive does not increase (Corollary 7.36), so \(\ee^{-\beta(x-x_{0})}U(x) \le U(x_{0}) = \alpha\) throughout, and \(u(x) \le U(x) \le \alpha\,\ee^{\beta(x-x_{0})}\).

Derivation. Derives Proposition 9.11. Both solutions satisfy the integral equation Equation (9.7) with their own initial values. Subtracting and using Equation (9.4),

This is Equation (9.10) for the continuous non-negative function \(u = \norm{\vect{Y}-\tilde{\vect{Y}}}\) with \(\alpha = \norm{\vect{Y}_{0}-\tilde{\vect{Y}}_{0}}\) and \(\beta = L\), so Lemma 9.10 gives Equation (9.12). Taking \(\alpha = 0\) recovers the uniqueness half of Theorem 9.8.

Derivation. Derives Theorem 9.13. Linearity of \(L\) makes \(\ker L\) a vector space and makes the difference of two solutions of \(L[y]=b\) a solution of \(L[y]=0\), which is the second statement. Fix \(x_{0} \in I\) and consider the evaluation map

which is linear. It is surjective because Corollary 9.9 produces a solution for every prescribed initial vector, and injective because that solution is unique: a \(y\) in \(\ker L\) with \(E(y) = \vect{0}\) agrees with the zero solution. An isomorphism onto \(\R^{n}\) forces \(\dim \ker L = n\).

Derivation. Derives Proposition 9.15. A determinant is multilinear in its rows, so \(\dd W/\dd x\) is a sum of \(n\) determinants, the \(i\)-th obtained by differentiating the \(i\)-th row of Equation (9.13). For \(i < n\) the differentiated row becomes equal to the row below it, so that determinant has two equal rows and vanishes. Only the last term survives, and in it the bottom row is \(\bigl(y_{1}^{(n)},\ldots,y_{n}^{(n)}\bigr)\). Each solution obeys \(y_{j}^{(n)} = -\sum_{i<n}\left(a_{i}/a_{n}\right)y_{j}^{(i)}\), so that row is a linear combination of the rows of Equation (9.13); subtracting the multiples of the rows carrying \(y^{(i)}\) for \(i \le n-2\), which changes no determinant, leaves \(-a_{n-1}/a_{n}\) times the row \(\bigl(y_{1}^{(n-1)},\ldots,y_{n}^{(n-1)}\bigr)\), that is \(-\left(a_{n-1}/a_{n}\right)W\). The resulting first-order linear equation integrates to the stated exponential, and an exponential never vanishes, so \(W(x)\) and \(W(x_{0})\) vanish together.

Derivation. Derives Proposition 9.16. If \(\sum_{j}c_{j}y_{j}\) vanishes identically with the \(c_{j}\) not all zero, differentiating \(n-1\) times shows that the columns of Equation (9.13) are linearly dependent at every \(x\), so \(W\) vanishes identically. Conversely, if \(W(x_{0}) = 0\) there is a non-zero vector \(\vect{c}\) annihilating the columns at \(x_{0}\), and \(y = \sum_{j}c_{j}y_{j}\) is then a solution with \(E(y) = \vect{0}\) in the notation of the proof of Theorem 9.13; by uniqueness \(y\) vanishes identically, so the \(y_{j}\) are dependent. Proposition 9.15 makes “at one point” and “at every point” the same statement. Finally \(n\) independent solutions inside an \(n\)-dimensional space (Theorem 9.13) are a basis of it.

Derivation. Derives Proposition 9.18. \(P\) is differentiable with \(P' = p\) (Theorem 7.42), so for any differentiable \(y\),

and \(\ee^{P}\) never vanishes. Hence \(y\) solves Equation (9.15) if and only if \(\bigl(\ee^{P}y\bigr)' = q\,\ee^{P}\), which by Theorem 7.43 holds if and only if \(\ee^{P(x)}y(x) - \ee^{P(x_{0})}y(x_{0}) = \int_{x_{0}}^{x}q\,\ee^{P}\); since \(P(x_{0}) = 0\) this is Equation (9.16) with \(C = y(x_{0})\). Every \(C\) occurs and no solution is missed, because Corollary 9.9 gives exactly one solution on \(I\) for each initial value.

Derivation. Derives Proposition 9.19. Set \(H(y) = \int_{y_{0}}^{y}\dd v/h(v)\) on \(J\) and \(G(x) = \int_{x_{0}}^{x}g(s)\,\dd s\) on \(I\); both are differentiable, with \(H' = 1/h\) and \(G' = g\) (Theorem 7.42). Since \(h\) is continuous and nowhere zero on the interval \(J\) it has one sign there (Theorem 7.23), so \(H'\) has one sign and \(H\) is strictly monotone (Corollary 7.36); it therefore maps \(J\) bijectively onto an interval \(H(J)\) containing \(0 = H(y_{0})\), and its inverse is differentiable with \((H^{-1})' = 1/H'\circ H^{-1} = h\circ H^{-1}\) (Proposition 7.32).

Let \(I_{0}\subseteq I\) be a neighbourhood of \(x_{0}\) on which \(G(x) \in H(J)\); such a neighbourhood exists because \(G\) is continuous and \(G(x_{0}) = H(y_{0}) = 0\) is an interior point of \(H(J)\), \(H\) being a strictly monotone continuous map and \(y_{0}\) interior to \(J\). On \(I_{0}\) define \(y = H^{-1}\circ G\). It takes the value \(H^{-1}(0) = y_{0}\) at \(x_{0}\), and by the chain rule (Proposition 7.31)

so it solves Equation (9.18). Conversely, if \(y\) is any solution with values in \(J\), then \(\dd\bigl[H(y(x)) - G(x)\bigr]/\dd x = y'/h(y) - g = 0\), so \(H(y(x)) - G(x)\) is constant (Corollary 7.36) and equal to its value \(0\) at \(x_{0}\); that is Equation (9.19), and \(H\) being injective determines \(y\) uniquely.

Derivation. Derives Proposition 9.22. Submultiplicativity gives \(\norm{x^{k}A^{k}/k!} \le \abs{x}^{k}\norm{A}^{k}/k! \le R^{k}\norm{A}^{k}/k!\) on \(\abs{x}\le R\), and those bounds sum to \(\ee^{R\norm{A}}\); by Lemma 9.6 the series converges absolutely and uniformly there, entry by entry. Each entry is a power series in \(x\) with infinite radius of convergence, so it may be differentiated term by term (Theorem 7.51), and \(\sum_{k\ge1}kx^{k-1}A^{k}/k! = A\sum_{k\ge0}x^{k}A^{k}/k!\), the factor \(A\) coming out on either side since it commutes with its own powers. That is Equation (9.22).

For the addition law with \(AB = BA\), expand the Cauchy product of the two absolutely convergent series (Proposition 7.49) and collect the terms of total degree \(m\): \(\sum_{j+k=m}A^{j}B^{k}/(j!\,k!) = (A+B)^{m}/m!\), the binomial theorem being available precisely because \(A\) and \(B\) commute. Taking \(B = yA\), which commutes with \(A\), gives \(\ee^{(x+y)A} = \ee^{xA}\ee^{yA}\), and \(y = -x\) identifies the inverse.

Derivation. Derives Theorem 9.23. Equation (9.22) makes Equation (9.24) a solution with the right initial value, and Corollary 9.9 says there is no other and that it lives on the whole line. For the inhomogeneous system, multiply by the integrating factor \(\ee^{-(x-x_{0})A}\) — the matrix analogue of Equation (9.17), legitimate because \(\ee^{-xA}\) commutes with \(A\) — to get \(\bigl(\ee^{-(x-x_{0})A}\vect{Y}\bigr)' = \ee^{-(x-x_{0})A}\vect{g}\), and integrate (Theorem 7.43). Multiplying back by \(\ee^{(x-x_{0})A}\) and using the addition law gives Equation (9.25).

Derivation. Derives Corollary 9.25. For an eigenvector, \(A^{k}\vect{v}_{r} = \lambda_{r}^{k}\vect{v}_{r}\), so term-by-term application of Equation (9.21) gives \(\ee^{xA}\vect{v}_{r} = \ee^{\lambda_{r}x}\vect{v}_{r}\). Expanding \(\vect{Y}_{0} = \sum_{r}c_{r}\vect{v}_{r}\) in the eigenbasis — a basis by hypothesis — and applying Equation (9.24) gives Equation (9.26). For the real form, conjugating \(A\vect{v} = \lambda\vect{v}\) with \(A\) real gives \(A\bar{\vect{v}} = \bar\lambda\bar{\vect{v}}\); the real and imaginary parts of the complex solution \(\ee^{\lambda x}\vect{v}\) are then themselves solutions, because \(A\) is real and the equation is linear, and expanding \(\ee^{(\alpha+\ii\beta)x}(\vect{a}+\ii\vect{b})\) with Euler's formula (Proposition 8.4) exhibits them as Equation (9.27).

Derivation. Derives Theorem 9.26. The resolvent. For \(\abs{\lambda} > \norm{A}\) the series \(\sum_{k\ge0}A^{k}\lambda^{-k-1}\) converges absolutely and uniformly, by Lemma 9.6 with \(M_{k} = \norm{A}^{k}\abs{\lambda}^{-k-1}\) and the geometric series Proposition 7.46; multiplying it by \(\lambda\identity - A\) telescopes to \(\identity\), so its sum is the resolvent

Solving \((\lambda\identity - A)\vect{X} = \vect{e}_{j}\) by Cramer's rule Equation (5.16) expresses the \((i,j)\) entry of \(R(\lambda)\) as a quotient \(D_{ij}(\lambda)/\chi(\lambda)\) in which \(D_{ij}\) is the determinant of a matrix whose entries are affine in \(\lambda\), hence a polynomial of degree at most \(n-1\). So each entry of \(R\) is a rational function of \(\lambda\) whose poles lie among the roots of \(\chi\), that of \(\lambda_{j}\) being of order at most \(m_{j}\).

The contour integral. Let \(\Gamma\) be the positively oriented circle \(\abs{\lambda} = \rho\) with \(\rho > \norm{A}\), so that Equation (9.29) holds on \(\Gamma\) and every root of \(\chi\) lies inside (an eigenvalue \(\lambda_{j}\) has \(\abs{\lambda_{j}} \le \norm{A}\), since an eigenvector would otherwise be lengthened by \(A\) beyond the bound its norm allows). Multiplying Equation (9.29) by \(\ee^{\lambda x}\), which is bounded on \(\Gamma\), preserves the uniform convergence, so the integral may be taken term by term; and \((2\pi\ii)^{-1}\oint_{\Gamma}\ee^{\lambda x}\lambda^{-k-1}\dd\lambda = x^{k}/k!\), the \(k\)-th Taylor coefficient of \(\ee^{\lambda x}\) at \(\lambda = 0\) (Theorem 8.17). Summing,

The residues. By the residue theorem Theorem 8.24 the right-hand side of Equation (9.30) is the sum over \(j\) of the residues of \(\ee^{\lambda x}R(\lambda)\) at \(\lambda_{j}\). Near \(\lambda_{j}\) the entry \((\lambda-\lambda_{j})^{m_{j}}R_{ik}(\lambda)\) is holomorphic and has a Taylor expansion \(\sum_{l\ge0}c_{l}(\lambda-\lambda_{j})^{l}\) (Theorem 8.20), while \(\ee^{\lambda x} = \ee^{\lambda_{j}x} \sum_{i\ge0}x^{i}(\lambda-\lambda_{j})^{i}/i!\). The residue is the coefficient of \((\lambda-\lambda_{j})^{-1}\) in the product (Theorem 8.21 and Definition 8.22), that is

a polynomial in \(x\) of degree at most \(m_{j}-1\) times \(\ee^{\lambda_{j}x}\). Summing over \(j\) gives Equation (9.28), and Equation (9.24) carries it to every solution. Replacing \(\ee^{\lambda x}\) throughout by \(\lambda^{N}\) — also holomorphic, with \((2\pi\ii)^{-1}\oint\lambda^{N-k-1}\dd\lambda = \delta_{k N}\) — gives \(A^{N}\) in place of \(\ee^{xA}\) and the stated form of its entries.

Derivation. Derives Corollary 9.27. By Equation (9.28) each entry of \(\ee^{xA}\) is a finite sum of terms \(x^{k}\ee^{\lambda_{j}x}\), whose modulus is \(x^{k}\ee^{x\Re\lambda_{j}}\). For \(\sigma > \sigma(A)\) the quotient \(x^{k}\ee^{x\Re\lambda_{j}}/\ee^{\sigma x} = x^{k}\ee^{-(\sigma-\Re\lambda_{j})x}\) tends to zero as \(x\to+\infty\) and is continuous on \([0,\infty)\), hence bounded there; summing the finitely many bounds gives Equation (9.31). If every \(\Re\lambda_{j}<0\), choose \(\sigma<0\) and every solution decays exponentially. Conversely an eigenvector for \(\lambda\) furnishes the solution \(\ee^{\lambda x}\vect{v}\) of Corollary 9.25, whose norm is \(\ee^{x\Re\lambda}\norm{\vect{v}}\): it does not tend to zero when \(\Re\lambda \ge 0\) and is unbounded when \(\Re\lambda>0\). When \(A\) has an eigenbasis, Equation (9.26) writes every solution as a fixed combination of the \(\ee^{\lambda_{r}x}\vect{v}_{r}\), each of norm \(\ee^{x\Re\lambda_{r}}\norm{\vect{v}_{r}} \le \norm{\vect{v}_{r}}\) for \(x\ge0\) when \(\Re\lambda_{r}\le0\); the triangle inequality bounds the sum. The statements about \(A^{N}\) are the same two arguments on the second half of Theorem 9.26, with \(\abs{\lambda_{j}}^{N}\) in place of \(\ee^{x\Re\lambda_{j}}\) and, in the diagonalizable case, with \(A^{N}\vect{v}_{r} = \lambda_{r}^{N}\vect{v}_{r}\).

Derivation. Derives Proposition 9.28. Write \(D\) for \(\dd/\dd x\). Polynomials in \(D\) with constant coefficients multiply exactly as polynomials in \(\lambda\) do, because \(D\) commutes with itself and with constants, so the factorization \(P(\lambda) = a_{n}\prod_{j}(\lambda-\lambda_{j})^{m_{j}}\) furnished by the fundamental theorem of algebra (Theorem 8.19) gives the operator identity \(L = a_{n}\prod_{j}(D-\lambda_{j})^{m_{j}}\), in which the factors may be applied in any order.

A single factor acts on the proposed solutions by

(Proposition 7.30), so \((D-\lambda)^{m}\) annihilates \(x^{\kappa}\ee^{\lambda x}\) for every \(\kappa\le m-1\). Applying the factors of \(L\) in an order that puts \((D-\lambda_{j})^{m_{j}}\) last shows every function Equation (9.33) solves \(L[y]=0\).

They are independent. Suppose \(\sum_{j}p_{j}(x)\ee^{\lambda_{j}x}\) vanishes identically, with \(p_{j}\) polynomials; we show every \(p_{j}\) is zero, by induction on the number \(p\) of distinct exponents. For \(p=1\), an exponential never vanishes, so \(p_{1}\) has infinitely many roots and is the zero polynomial. For \(p>1\), multiply by \(\ee^{-\lambda_{p}x}\) and differentiate \(\deg p_{p}+1\) times. The last term dies; each other term \(p_{j}\ee^{\mu_{j}x}\), with \(\mu_{j} = \lambda_{j}-\lambda_{p}\neq0\), becomes \(q_{j}\ee^{\mu_{j}x}\) with \(\deg q_{j} = \deg p_{j}\), since each differentiation multiplies the leading coefficient by \(\mu_{j}\neq0\). The inductive hypothesis makes every \(q_{j}\), hence every \(p_{j}\) with \(j<p\), zero; and then \(p_{p}\) is zero by the case \(p=1\).

There are \(\sum_{j}m_{j} = n\) of them, and the solution space has dimension \(n\) (Theorem 9.13), so they are a basis. The real form of a conjugate pair is obtained as in Corollary 9.25, by taking the real and imaginary parts of \(\ee^{(\alpha+\ii\beta)x}\) (Proposition 8.4); they are independent because their Wronskian is \(\beta\,\ee^{2\alpha x}\neq0\) (Proposition 9.16).

Derivation. Derives Proposition 9.29. The characteristic polynomial of a \(2\times2\) matrix is \(\lambda^{2}-\tau\lambda+\Delta\) (Theorem 5.71), whose roots are Equation (9.34); they multiply to \(\Delta\) and add to \(\tau\). If \(D>0\) the roots are real and distinct: they have opposite signs when \(\Delta<0\) (a saddle, one growing and one decaying direction) and the common sign of \(\tau\) when \(\Delta>0\) (a node). If \(D<0\) they are a conjugate pair \(\tau/2\pm\ii\beta\) with \(\beta = \sqrt{-D}/2 \neq 0\), and Equation (9.27) shows every solution to be a rotation of period \(2\pi/\beta\) scaled by \(\ee^{\tau x/2}\): a focus spiralling in for \(\tau<0\) and out for \(\tau>0\), and for \(\tau=0\) a centre, every orbit of which is closed and periodic. If \(D=0\) the eigenvalue \(\tau/2\) is double and Equation (9.28) permits a factor of \(x\), present exactly when \(A\) is not a multiple of the identity: a degenerate node, or a star node when it is. The stability statements are Corollary 9.27 read through \(\Re\lambda_{\pm}\), both of which are negative precisely when their sum \(\tau\) is negative and their product \(\Delta\) positive.

Derivation. Derives Proposition 9.31. Put \(\vect{Z} = \vect{Y}-\vect{Y}_{\ast}\) and \(\vect{N}(\vect{Z}) = \vect{F}(\vect{Y}_{\ast}+\vect{Z}) - A\vect{Z}\), so that \(\vect{Z}' = A\vect{Z} + \vect{N}(\vect{Z})\). Because \(\vect{F}\) is \(C^{1}\) and \(A\) is its derivative at the equilibrium, \(\norm{\vect{N}(\vect{Z})}/\norm{\vect{Z}} \to 0\) as \(\vect{Z}\to\vect{0}\) (Theorem 7.100): given any \(\eta>0\) there is a \(\delta>0\) with \(\norm{\vect{N}(\vect{Z})}\le\eta\norm{\vect{Z}}\) for \(\norm{\vect{Z}}\le\delta\).

Choose \(\sigma\) with \(\sigma(A)<\sigma<0\) and let \(C\) be the constant of Equation (9.31). Fix \(\eta = -\sigma/(2C)\) and the corresponding \(\delta\), and treat \(\vect{N}\) as an inhomogeneity in Equation (9.25): as long as the solution stays within \(\delta\),

Multiplying by \(\ee^{-\sigma x}\) and writing \(u(x) = \ee^{-\sigma x}\norm{\vect{Z}(x)}\) turns this into \(u(x) \le C\norm{\vect{Z}(0)} + C\eta\int_{0}^{x}u\), so Lemma 9.10 gives \(u(x)\le C\norm{\vect{Z}(0)}\ee^{C\eta x}\), that is

by the choice of \(\eta\). Taking \(\varepsilon = \delta/C\) makes the right-hand side at most \(\delta\) for every \(x\ge0\), so the solution never leaves the ball on which the estimate was assumed and the bound holds for all \(x\ge0\). This is Equation (9.35) with \(\mu = -\sigma/2 > 0\).

Derivation. Derives Proposition 9.36. Suppose a closed orbit \(\gamma\) lay in \(\Omega\), and let \(S\) be the region it bounds, contained in \(\Omega\) because \(\Omega\) is simply connected. Green's theorem Theorem 7.131 equates the integral of Equation (9.36) over \(S\) with the circulation \(\oint_{\gamma}\varphi\,(F_{1}\,\dd y_{2} - F_{2}\,\dd y_{1})\). On the orbit \(\dd y_{1} = F_{1}\,\dd x\) and \(\dd y_{2} = F_{2}\,\dd x\), so the integrand is \(\varphi\,(F_{1}F_{2}-F_{2}F_{1})\,\dd x = 0\) and the circulation vanishes. But the area integral of a function of one strict sign over a region of positive area cannot vanish — a contradiction.

Derivation. Derives Proposition 9.38. This is the computation of Proposition 9.15 carried out on the columns of \(\Phi\) instead of on a Wronskian. A determinant is multilinear in its rows, so \(\dd(\det\Phi)/\dd x\) is the sum over \(i\) of the determinant of \(\Phi\) with its \(i\)-th row differentiated. By Equation (9.37) that row is \(\sum_{k}A_{ik}\Phi_{k\cdot}\), a combination of the rows of \(\Phi\); subtracting the multiples of the other rows, which changes no determinant, leaves \(A_{ii}\det\Phi\). Summing over \(i\) gives \(\dd(\det\Phi)/\dd x = (\tr A)\det\Phi\), a scalar linear equation whose solution with \(\det\Phi(x_{0}) = 1\) is Equation (9.38) by Proposition 9.18. An exponential never vanishes.

Derivation. Derives Theorem 9.39. (i) Put \(\Psi(x) = \Phi(x+T)\). Then \(\Psi' = A(x+T)\Psi = A(x)\Psi\) by periodicity, so \(\Psi\) solves Equation (9.37) column by column; and \(\Phi(x)M\) solves it too, being a fixed linear combination of solutions. The two agree at \(x=0\), where both equal \(M\), so they agree everywhere by the uniqueness in Corollary 9.9. Iterating gives the statement for \(NT\).

(ii) is Equation (9.38) at \(x=T\) together with the fact that the determinant is the product of the eigenvalues (Theorem 5.71, comparing the constant terms of the characteristic polynomial).

(iii) Let \(M\vect{w} = \rho\vect{w}\) with \(\vect{w}\neq\vect{0}\) and put \(\vect{Y}(x) = \Phi(x)\vect{w}\). By (i), \(\vect{Y}(x+T) = \Phi(x)M\vect{w} = \rho\,\vect{Y}(x)\). Since \(\rho \neq 0\) by (ii), the complex exponential takes the value \(\rho\) (Proposition 8.4: write \(\rho = \abs{\rho}\ee^{\ii\theta}\) and take \(\mu T = \ln\abs{\rho} + \ii\theta\)). Setting \(\vect{P}(x) = \ee^{-\mu x}\vect{Y}(x)\) gives \(\vect{P}(x+T) = \ee^{-\mu T}\ee^{-\mu x}\rho\,\vect{Y}(x) = \vect{P}(x)\), which is Equation (9.39).

(iv) By (i), \(\vect{Y}(x+NT) = \Phi(x)M^{N}\vect{Y}(0)\). The matrix \(\Phi\) is continuous, hence bounded on the compact interval \([0,T]\) (Theorem 7.24), and \(\Phi(x)^{-1}\) is bounded there too by Equation (9.38). So the solution tends to zero, or is bounded, exactly as the sequence \(M^{N}\vect{Y}(0)\) does, and the statements about the powers \(A^{N}\) in Corollary 9.27 settle that by the moduli of the multipliers: \(M^{N}\to0\) when every \(\abs{\rho_{i}}<1\), while an eigenvector for a multiplier of modulus greater than one gives \(M^{N}\vect{w} = \rho^{N}\vect{w}\), unbounded.

Derivation. Derives Corollary 9.41. In the system for \(\vect{Y} = (y,y')\) the matrix is \(A = \begin{pmatrix} 0 & 1\\ -Q & 0\end{pmatrix}\), of zero trace, so Equation (9.38) gives \(\det M = 1\) and the characteristic polynomial of the \(2\times2\) matrix \(M\) is Equation (9.41) (Proposition 9.29, the same identity read for \(M\)). Its roots are \(\rho_{\pm} = \bigl[\Delta_{\mathrm{H}} \pm\sqrt{\Delta_{\mathrm{H}}^{2}-4}\,\bigr]/2\).

If \(\abs{\Delta_{\mathrm{H}}}<2\) the discriminant is negative, the roots are complex conjugates of product \(1\), hence of modulus \(1\), and distinct; \(M\) then has two independent eigenvectors (Proposition 5.75), so the powers \(M^{N}\) stay bounded by the last clause of Corollary 9.27, and by the argument of Theorem 9.39(iv) every solution is bounded. If \(\abs{\Delta_{\mathrm{H}}}>2\) the roots are real and distinct with product \(1\), so one has modulus greater than \(1\) and Theorem 9.39(iv) gives an unbounded solution; by Equation (9.39) it is \(\ee^{\mu x}\) times a periodic function, with \(\ee^{\mu T} = \rho\). If \(\abs{\Delta_{\mathrm{H}}}=2\) the double root is \(\rho = \Delta_{\mathrm{H}}/2 = \pm1\), and the eigenvector of \(M\) belonging to it furnishes, by Theorem 9.39(iii), a solution with \(y(x+T) = \pm y(x)\).

Derivation. Derives Proposition 9.43. At \(h=0\) the equation is \(y''+ay=0\) with constant coefficients, whose fundamental matrix over one period \(T=\pi\) is computed from Theorem 9.23. For \(a>0\) the solutions are \(\cos(\sqrt{a}\,x)\) and \(\sin(\sqrt{a}\,x)/\sqrt{a}\), so

Hence \(\abs{\Delta_{\mathrm{H}}}<2\), the stable case of Corollary 9.41, at every \(a>0\) except where \(\cos(\pi\sqrt{a}) = \pm1\), that is where \(\pi\sqrt{a} = n\pi\), that is at \(a = n^{2}\) with \(n\ge1\); there \(\abs{\Delta_{\mathrm{H}}}=2\) and the second solution grows linearly, so those points belong to the closure of the unstable set. For \(a=0\) the solutions are \(1\) and \(x\), giving \(\Delta_{\mathrm{H}} = 2\) with an unbounded solution, and for \(a<0\) they are real exponentials with \(\Delta_{\mathrm{H}} = 2\cosh(\pi\sqrt{-a}) > 2\), the unstable case outright. So on the line \(h=0\) the unstable set is the ray \(a\le0\) together with the isolated points \(a = n^{2}\), \(n\ge1\), which is the statement.

Derivation. Derives Proposition 9.48. Multiply Equation (9.47) by \(\overline{\phi_j(x)}\) and integrate over \(I\):

where the orthonormality relation Equation (9.46) collapsed the sum.

Derivation. Derives Proposition 9.50. Applying Equation (9.48) to the product and substituting the two series,

and the remaining integral is \(T_{ijk}\) by Equation (9.49).

Derivation. Derives Proposition 9.52. Every integral carries the measure \(\dd x/(2\pi)\) (Remark 9.91), which is also what puts the factor into Equation (9.111), so the triple overlap to be computed is

by Equation (9.109) applied to the integer \(j+k-n\). Inserting this in Equation (9.50) collapses the double sum to the single sum over \(j = m\), \(k = n-m\), which is Equation (9.51). Absolute convergence, and with it the right to rearrange the double series into that single one, is the Cauchy product of two absolutely convergent series (Proposition 7.49).

The other reading of Remark 9.91 gives the same answer, as it must. For the set \(\phi_n = \ee^{\ii n x}/\sqrt{2\pi}\) with plain \(\dd x\) the same integral yields \(T_{njk} = \delta_{n,\,j+k}/\sqrt{2\pi}\) while every coefficient is multiplied by \(\sqrt{2\pi}\), and the three factors cancel in Equation (9.50), leaving Equation (9.51) unchanged.

Derivation. Derives Equation (9.52). Mean-square convergence permits the series to be integrated term by term against its own conjugate:

the orthonormality relation Equation (9.46) collapsing the double sum to its diagonal.

Derivation. Derives Lemma 9.56. The terms carrying \(q\) cancel, and by the Leibniz rule Equation (7.16),

in which the two products of first derivatives cancel, leaving Equation (9.56).

Derivation. Derives Theorem 9.57. By hypothesis \(L[u] = -\lambda_{u}^{2}\,r\,u\) and \(L[v] = -\lambda_{v}^{2}\,r\,v\), so that \(u\,L[v] - v\,L[u] = \bigl(\lambda_{u}^{2}-\lambda_{v}^{2}\bigr)\,r\,u\,v\). Integrating Equation (9.56) over \([a,b]\) therefore gives

which is zero by Equation (9.57). Dividing by the non-zero factor \(\lambda_{u}^{2}-\lambda_{v}^{2}\) gives Equation (9.58).

Derivation. Derives Proposition 9.60. Differentiating Equation (9.62) twice,

Substituting into Equation (9.60) and cancelling the nowhere-vanishing factor \(\psi\) leaves

and multiplication by \(-\hbar^{2}\) gives Equation (9.63).

Derivation. Derives Proposition 9.61. Insert Equation (9.64) in Equation (9.63). The squared term contributes \((S_{0}')^{2} + 2\hbar S_{0}'S_{1}' + \hbar^{2}\bigl[(S_{1}')^{2} + 2S_{0}'S_{2}'\bigr] + \cdots\), and the second term contributes \(-\ii\hbar S_{0}'' - \ii\hbar^{2}S_{1}'' - \cdots\); the right-hand side \(P^{2}\) is of order \(\hbar^{0}\). Equating coefficients of \(\hbar^{0}\), \(\hbar^{1}\) and \(\hbar^{2}\) gives Equations (9.65) to (9.67). The eikonal equation is solved by quadrature, with the two signs corresponding to the two directions of classical motion, which is the first equality of Equation (9.68). Dividing Equation (9.66) by \(2S_{0}'\) turns it into

and \(\abs{S_{0}'} = P\) by Equation (9.65), so integrating gives the second equality.

Derivation. Derives Corollary 9.62. By Equation (9.62) and Equation (9.68),

because \(\ii S_{1} = \ii\cdot\tfrac{\ii}{2}\ln P = -\tfrac12\ln P\), and the second factor is \(P^{-1/2}\). Linear superposition of the two signs gives Equation (9.69). In a forbidden region \(P = \ii\kappa\), the two exponents become real, and absorbing the constant phase of \(P^{-1/2}\) into the coefficients gives Equation (9.70).

Derivation. Derives Proposition 9.63. Write \(\psi = C_{+}P^{-1/2}\ee^{\ii\theta}\) with \(\theta(x) = \hbar^{-1}\int^{x}P\,\dd x'\), so that \(\dd\theta/\dd x = P/\hbar\). Then

the first term inside the bracket being real because \(P\) is real in the allowed region. Taking the imaginary part and multiplying by \(\hbar/m\) gives Equation (9.71). Had the amplitude been anything other than \(P^{-1/2}\) the bracket would have carried an \(x\)-dependent imaginary part and the current would not have been constant, which for a stationary state on a line is impossible.

Derivation. Derives Proposition 9.65. By Equation (9.68), \(\hbar\,S_{1}' = \ii\hbar P'/(2P)\) and \(S_{0}' = \pm P\), so

the last equality because \(\dd\bar\lambda/\dd x = -\hbar P'/P^{2}\) from Equation (9.72). Differentiating \(P^{2} = 2m(E-V)\) gives \(2PP' = -2mV'\), hence \(P' = -mV'/P\) and \(\hbar\abs{P'}/P^{2} = \hbar m\abs{V'}/P^{3}\).

Derivation. Derives Proposition 9.69. The integrand is entire in \(t\) and decays superexponentially as \(\abs{t}\to\infty\) in each of the three open sectors where \(\Re\bigl(t^{3}\bigr) < 0\), namely those centred on \(\arg t = \pi/3\), \(\arg t = \pi\) and \(\arg t = -\pi/3\). Both ends of \(C\) lie in such sectors, so Equation (9.77) converges and may be differentiated under the integral sign. Since

one has

the bracket vanishing because both ends of \(C\) lie in decay sectors. If \(\omega^{3} = 1\) then \(w(z) = \mathrm{Ai}(\omega z)\) satisfies \(\dd^{2}w/\dd z^{2} = \omega^{2}\,\mathrm{Ai}''(\omega z) = \omega^{3}\,z\,\mathrm{Ai}(\omega z) = z\,w\), so each term of Equation (9.78) solves the equation and so does their sum. Reflection: conjugating Equation (9.77) maps \(C\) onto itself with reversed orientation and sends the integrand to \(\exp\bigl[t^{3}/3 - \bar z t\bigr]\), while \(\overline{1/(2\pi\ii)} = -1/(2\pi\ii)\); the two sign changes cancel and \(\overline{\mathrm{Ai}(z)} = \mathrm{Ai}(\bar z)\). Hence \(\mathrm{Ai}\) is real on the real axis, and for real \(z\) the two terms of Equation (9.78) are complex conjugates of each other, so \(\mathrm{Bi}\) is real there too. Independence follows from Theorem 9.70 below: one solution decays and the other grows as \(z\to+\infty\), so neither is a multiple of the other.

Derivation. Derives Theorem 9.70. Write the exponent of Equation (9.77) as \(\varphi(t) = t^{3}/3 - z\,t\), so that \(\varphi'(t) = t^{2}-z\) and \(\varphi''(t) = 2t\); the saddle points are \(t = \pm\sqrt{z}\). By Cauchy's theorem and the deformation corollary (Theorem 8.12 and Corollary 8.15) the contour may be moved at will, the integrand being entire, provided its ends stay in the decay sectors. The method of steepest descent chooses the deformation on which \(\Im\varphi\) is constant: the integrand then does not oscillate, its modulus has a sharp maximum at the saddle, and the whole contribution comes from a neighbourhood of it whose width shrinks as \(\abs{\varphi''}^{-1/2}\).

Positive argument. Let \(z = x > 0\). The two saddles are at \(t = \pm\sqrt{x}\), with \(\varphi(\sqrt{x}) = -\tfrac{2}{3}x^{3/2}\) and \(\varphi(-\sqrt{x}) = +\tfrac{2}{3}x^{3/2}\). The ends of \(C\) lie in the two sectors adjacent to the positive real axis, so \(C\) is deformed through the saddle at \(t = +\sqrt{x}\) alone. There \(\varphi'' = 2\sqrt{x}\) is real and positive, so putting \(t = \sqrt{x} + \ii u\) with \(u\) real makes \(\tfrac12\varphi''\,(t-\sqrt{x})^{2} = -\sqrt{x}\,u^{2}\): the vertical line through the saddle is the path of steepest descent. On it \(\dd t = \ii\,\dd u\), and

the first member of Equation (9.79). Nothing in this computation used \(z\) real: the same deformation is available, and the same result holds, for every \(z\) with \(\abs{\arg z} < \pi\).

Negative argument. Let \(z = -x\) with \(x > 0\), so that \(\varphi(t) = t^{3}/3 + x\,t\) and the saddles sit at \(t = \pm\ii\sqrt{x}\) with \(\varphi(\pm\ii\sqrt{x}) = \pm\ii\,\tfrac{2}{3}x^{3/2}\), purely imaginary: neither dominates the other, and both must be kept. The bookkeeping is topological. Call \(V_{1}\), \(V_{2}\), \(V_{3}\) the decay sectors around \(\arg t = \pi/3\), \(\pi\) and \(-\pi/3\); the contour runs from \(V_{3}\) to \(V_{1}\). At a saddle \(s\) the steepest descent direction \(\ee^{\ii\theta}\) is fixed by requiring \(\tfrac12\varphi''(s)\,\ee^{2\ii\theta}\) to be real and negative. At \(s = -\ii\sqrt{x}\), where \(\varphi'' = 2\sqrt{x}\,\ee^{-\ii\pi/2}\), this gives \(\theta = -\pi/4\) or \(\theta = 3\pi/4\), so that descent path joins \(V_{3}\) to \(V_{2}\); at \(s = +\ii\sqrt{x}\), where \(\varphi'' = 2\sqrt{x}\,\ee^{\ii\pi/2}\), it gives \(\theta = \pi/4\) or \(\theta = 5\pi/4\), joining \(V_{2}\) to \(V_{1}\). Their concatenation runs from \(V_{3}\) to \(V_{1}\) and is therefore a legitimate deformation of \(C\). Each saddle contributes \(\ee^{\varphi(s)}\,\ee^{\ii\theta}\sqrt{\pi/\sqrt{x}}\) with \(\theta\) the direction in which it is traversed — \(3\pi/4\) at the lower saddle and \(\pi/4\) at the upper one. Writing \(\zeta = \tfrac{2}{3}x^{3/2}\),

and \(\cos(\zeta-\pi/4) = \sin(\zeta+\pi/4)\), which is the second member of Equation (9.79).

The second solution. Equation (9.78) reduces both members of Equation (9.80) to the result already obtained. For \(z = x > 0\) the two arguments are \(x\,\ee^{\pm2\pi\ii/3}\), inside the sector \(\abs{\arg z} < \pi\) where the first expansion holds; there \(\bigl(x\ee^{\pm2\pi\ii/3}\bigr)^{3/2} = x^{3/2}\ee^{\pm\ii\pi} = -x^{3/2}\) and \(\bigl(x\ee^{\pm2\pi\ii/3}\bigr)^{1/4} = x^{1/4}\ee^{\pm\ii\pi/6}\), so each term of Equation (9.78) equals \(\ee^{\pm\ii\pi/6}\,\ee^{+\frac{2}{3}x^{3/2}} \bigm/ \bigl(2\sqrt{\pi}\,x^{1/4}\,\ee^{\pm\ii\pi/6}\bigr)\): the phases cancel, the two terms are equal, and their sum is the first member of Equation (9.80). For \(z = -x\) the two arguments are \(x\,\ee^{\mp\ii\pi/3}\), again inside the sector, and now \(\bigl(x\ee^{\mp\ii\pi/3}\bigr)^{3/2} = \mp\ii\,x^{3/2}\) and \(\bigl(x\ee^{\mp\ii\pi/3}\bigr)^{1/4} = x^{1/4}\ee^{\mp\ii\pi/12}\), so the two terms are \(\ee^{\pm\ii\pi/4}\,\ee^{\pm\ii\zeta} \bigm/\bigl(2\sqrt{\pi}\,x^{1/4}\bigr)\) and their sum is \(\cos(\zeta+\pi/4)\bigm/\bigl(\sqrt{\pi}\,x^{1/4}\bigr)\), the second member of Equation (9.80).

Derivation. Derives Theorem 9.72. Work with the linearised equation, for which the exact solution is a combination of \(\mathrm{Ai}(z)\) and \(\mathrm{Bi}(z)\), and evaluate the two sides of Equation (9.81) on it. For \(x > a\), that is \(z > 0\), Equation (9.61) gives \(\kappa(x) = \sqrt{2mF(x-a)} = \sqrt{2mF\ell}\;z^{1/2}\), hence

because \(\ell^{3/2} = \hbar/\sqrt{2mF}\) by Equation (9.74). So on the forbidden side the decaying semiclassical form is \(C\,(2mF\ell)^{-1/4}z^{-1/4}\ee^{-\frac{2}{3}z^{3/2}}\), which by Equation (9.79) is \(2\sqrt{\pi}\,C\,(2mF\ell)^{-1/4}\) times \(\mathrm{Ai}(z)\). For \(x < a\) put \(z = -u\) with \(u > 0\); then \(P(x) = \sqrt{2mF\ell}\;u^{1/2}\) and, by the same computation, \(\hbar^{-1}\int_{x}^{a}P\,\dd x' = \tfrac{2}{3}u^{3/2}\). The second relation of Equation (9.79) therefore gives

which is Equation (9.81). The factor \(2\) is nothing but the ratio of the two prefactors in Equation (9.79), and the phase \(\pi/4\) is the constant carried across by the same pair. Repeating the computation with \(\mathrm{Bi}\) in place of \(\mathrm{Ai}\), whose two prefactors in Equation (9.80) are equal, gives Equation (9.82) with no factor of \(2\). The case \(F<0\) is the reflection \(x \mapsto 2a - x\), which leaves Equation (9.60) invariant.

Derivation. Derives Proposition 9.74. Put \(s = x-a\) and \(k = \sqrt{2mF_{\mu}}/\hbar\), a positive constant. Near \(a\), Equation (9.60) reads

and for \(s>0\) the substitution \(\zeta = 2k\,s^{q}/(\mu+2)\) with \(q = (\mu+2)/2\), together with \(\psi = \sqrt{s}\,\chi(\zeta)\), turns it by a direct computation into the modified Bessel equation of order \(\nu = 1/(\mu+2)\). A basis of local solutions is therefore \(\sqrt{s}\,I_{\pm\nu}(\zeta)\), and the decaying combination is \(\sqrt{s}\,\bigl[I_{-\nu}(\zeta) - I_{\nu}(\zeta)\bigr]\), proportional to \(\sqrt{s}\,K_{\nu}(\zeta)\). Note that \(\zeta = \hbar^{-1}\int_{a}^{x}\kappa\,\dd x'\) exactly for the local potential, since \(\kappa = \hbar k\,s^{\mu/2}\) there.

Continue to the allowed side by rotating \(s = u\,\ee^{\ii\pi}\) with \(u = a - x > 0\). Then \(\zeta = w\,\ee^{\ii\pi q}\) with \(w = 2k\,u^{q}/(\mu+2) = \hbar^{-1}\int_{x}^{a}P\,\dd x'\). In the power series

the rotation multiplies the \(j\)-th term by \(\ee^{2\pi\ii q j} = \ee^{\ii\pi(\mu+2)j} = (-1)^{j}\), because \(\mu+2\) is odd; and an alternating sign is exactly what turns the modified Bessel series into the ordinary one. Hence \(I_{\sigma}\bigl(w\,\ee^{\ii\pi q}\bigr) = \ee^{\ii\pi q\sigma}\,J_{\sigma}(w)\). With \(q\nu = \tfrac12\) and \(\sqrt{s} = \ee^{\ii\pi/2}\sqrt{u}\) the two basis solutions continue to \(\ee^{\ii\pi/2}\ee^{\pm\ii\pi/2}\sqrt{u}\,J_{\pm\nu}(w)\), that is to \(-\sqrt{u}\,J_{\nu}(w)\) and \(+\sqrt{u}\,J_{-\nu}(w)\), so the decaying solution continues to \(\sqrt{u}\,\bigl[J_{-\nu}(w) + J_{\nu}(w)\bigr]\). This much is exact.

It remains to read off the two amplitudes, and for that one classical fact is imported: the large-argument behaviour of the Bessel functions, \(K_{\nu}(\zeta) \sim \sqrt{\pi/(2\zeta)}\;\ee^{-\zeta}\) and \(J_{\sigma}(w) \sim \sqrt{2/(\pi w)}\, \cos\left(w - \sigma\pi/2 - \pi/4\right)\) [Morse:1953]. The prefactors \(\sqrt{s/\zeta}\) and \(\sqrt{u/w}\) are the same function of \(\abs{s}\) with the same constant in front, and each reproduces the semiclassical amplitude, \(\kappa^{-1/2}\) on one side and \(P^{-1/2}\) on the other, so only the pure numbers matter. On the forbidden side \(I_{-\nu} - I_{\nu} = \left(2\sin\nu\pi/\pi\right)K_{\nu}\) leaves the number \(\left(2\sin\nu\pi/\pi\right)\sqrt{\pi/2} = \sqrt{2/\pi}\,\sin\nu\pi\). On the allowed side

the two phase shifts \(\mp\nu\pi/2\) averaging away and leaving the universal \(-\pi/4\). The ratio of the two numbers is

which with \(\nu = 1/(\mu+2)\) is the factor in Equation (9.84); and \(\cos(w-\pi/4) = \sin(w+\pi/4)\) completes it.

Derivation. Derives Theorem 9.76. Beyond the barrier, for \(x > b\), impose a purely outgoing wave and write it with the phase measured from \(b\),

with \(\Theta(x) = \hbar^{-1}\int_{b}^{x}P\,\dd x' + \pi/4\). At \(b\) the forbidden region lies to the left, so Theorem 9.72 is used in its reflected form. The direction has to be said carefully, because “growing towards \(b\)” and “growing away from \(b\)” name opposite solutions: measured into the barrier, the way Equations (9.81) and (9.82) measure it, the \(\cos\Theta\) part comes from the exponential growing away from \(b\) and the \(\sin\Theta\) part from the one decaying away from it, with the coefficients read off Equation (9.82) and Equation (9.81) respectively. Hence inside the barrier

The first of these two readings is in the safe direction of Remark 9.73; the second is not, and it is the sole source of the order-unity ambiguity recorded in Remark 9.77 — it is the term that carries the coefficient \(\tfrac14\) into Equation (9.86), and it does not affect the exponent.

Now re-express the exponents about the other turning point using \(\int_{x}^{b}\kappa = \hbar\gamma - \int_{a}^{x}\kappa\), which turns the first term into \(\mathcal{A}\ee^{\gamma}\) times the exponential decaying away from \(a\) and the second into \((\ii \mathcal{A}/2)\ee^{-\gamma}\) times the one growing away from \(a\). At \(a\) the allowed region lies to the left, which is exactly the configuration of Theorem 9.72, so with \(\Xi(x) = \hbar^{-1}\int_{x}^{a}P\,\dd x' + \pi/4\),

Writing \(\sin\Xi = (\ee^{\ii\Xi}-\ee^{-\ii\Xi})/(2\ii)\) and \(\cos\Xi = (\ee^{\ii\Xi}+\ee^{-\ii\Xi})/2\) and collecting,

Since \(\dd\Xi/\dd x = -P/\hbar\), the term in \(\ee^{-\ii\Xi}\) advances in phase with increasing \(x\) and is the incident wave, while \(\ee^{\ii\Xi}\) is the reflected one. The amplitude \(P^{-1/2}\) has already made \(\abs{\,\cdot\,}^{2}\) proportional to the current, by Proposition 9.63, so the transmission coefficient is the ratio of the squared coefficients of the transmitted and incident waves,

which is Equation (9.86). Expanding, \(T = \ee^{-2\gamma}\bigl(1+\tfrac14\ee^{-2\gamma}\bigr)^{-2} = \ee^{-2\gamma}\bigl[1 - \tfrac12\ee^{-2\gamma} + \cdots\bigr]\).

Derivation. Derives Theorem 9.78. Decay in the left forbidden region, \(x < a\), is the configuration of Theorem 9.72 reflected, and connects to

inside the well. Decay in the right forbidden region, \(x > b\), connects by Equation (9.81) to

These are two expressions for the same function, so they must agree up to sign for every \(x\) in \((a,b)\). Write \(\theta(x) = \hbar^{-1}\int_{a}^{x}P\,\dd x'\) and \(\Phi = \hbar^{-1}\int_{a}^{b}P\,\dd x'\), so that \(\hbar^{-1}\int_{x}^{b}P\,\dd x' = \Phi - \theta\), and expand

For this to be a constant multiple of \(\sin(\theta + \pi/4)\) for all \(\theta\), the coefficient of the independent function \(\cos(\theta+\pi/4)\) must vanish, that is \(\cos\Phi = 0\), whence \(\Phi = (n + \tfrac12)\pi\) with \(n\) a non-negative integer, which is Equation (9.88). The remaining coefficient is then \(-\cos(\Phi+\pi/2) = \sin\Phi = (-1)^{n}\), so \(C' = (-1)^{n}C\) and the two expressions do agree. Doubling and multiplying by \(\hbar\) gives \(\oint P\,\dd x = (2n+1)\pi\hbar = (n+\tfrac12)h\), which is Equation (9.89).

Derivation. Derives Theorem 9.83. From Equation (9.96), \(\dd/\dd r = r^{-1}\dd/\dd\varrho\) with \(r = r_{0}\ee^{\varrho}\), so

the cross terms in \(\dd w/\dd\varrho\) cancelling in the second derivative, which is what the factor \(\ee^{\varrho/2}\) in Equation (9.96) was chosen to arrange. Substituting into Equation (9.95) written as

and multiplying through by \(r_{0}^{2}\,\ee^{3\varrho/2}\) gives

because \(r_{0}^{2}\ee^{2\varrho} = r^{2}\). The two constant terms combine as \(n(n+1) + \tfrac14 = \left(n+\tfrac12\right)^{2}\), which is Equation (9.97). For Equation (9.98), \(\dd\varrho = \dd r/r\), so

Both terms of \(Q^{2}\) carry the square of \(\mathrm{J}\,\mathrm{s}\) — the first because \(2m\left(E-V\right)r^{2}\) is a mass times an energy times an area, the second manifestly — so \(Q/\hbar\) is dimensionless, as an exponent per unit of \(\varrho\) must be; the arbitrary \(r_{0}\) has cancelled out of both Equation (9.97) and Equation (9.98), as it must, since shifting it only translates \(\varrho\).

Derivation. Derives Proposition 9.87. Under \(x = x_{c}+\delta\xi\) the three terms of Equation (9.101) carry the factors \(\varepsilon/\delta^{2}\), \(1/\delta\) and \(1\). Suppose the first is retained. If it balances the third, \(\varepsilon/\delta^{2}\sim1\) and \(\delta\sim\sqrt{\varepsilon}\), whereupon the second term is of order \(\varepsilon^{-1/2}\) and dominates both retained terms — so the balance is inconsistent. If it balances the second, \(\varepsilon/\delta^{2}\sim1/\delta\), that is \(\delta\sim\varepsilon\), and the third term, of order \(1\), is smaller by a factor \(\varepsilon\) than the two retained ones: consistent. That is (i), and substituting \(\delta=\varepsilon\) and \(a(x_{c}+\varepsilon\xi)\to a(x_{c})\) gives (ii), whose solutions are read off Theorem 9.23 for the constant-coefficient equation — or directly from Proposition 9.18 applied to \(\dd Y/\dd\xi\).

For (iii): at \(x_{c}=0\) the interval is \(\xi\ge0\) and \(\ee^{-a(0)\xi}\to0\) as \(\xi\to\infty\), so the inner solution settles to a constant and Equation (9.99) can be imposed. At \(x_{c}=1\) the interval is \(\xi\le0\) and \(\ee^{-a(1)\xi}\to\infty\) as \(\xi\to-\infty\): the inner solution grows without bound where it should match, so no matching is possible and there is no layer there. The reduced equation is of first order and therefore accepts exactly one boundary condition, which must then be the one at \(x=1\).

Derivation. Derives Proposition 9.88. The equation has constant coefficients, so by Proposition 9.28 its solutions are combinations of \(\ee^{\lambda_{\pm}x}\) with \(\lambda_{\pm} = \bigl[-1\pm\sqrt{1-4\varepsilon}\bigr]/(2\varepsilon)\), real and distinct for \(\varepsilon<1/4\). Taylor's theorem (Theorem 7.38) applied to \(\sqrt{1-u}\) at \(u=4\varepsilon\) gives \(\abs{\sqrt{1-4\varepsilon}-(1-2\varepsilon)} \le c_{0}\varepsilon^{2}\) on \((0,1/8]\), whence

In particular \(\lambda_{-}\le-1/(2\varepsilon)\), so \(\eta = \ee^{\lambda_{-}}\le\ee^{-1/(2\varepsilon)}\) is smaller than any power of \(\varepsilon\).

Write \(y = C_{+}\ee^{\lambda_{+}x} + C_{-}\ee^{\lambda_{-}x}\). The boundary conditions give \(C_{+}+C_{-}=A\) and \(C_{+}\ee^{\lambda_{+}}+C_{-}\eta = B\), so \(C_{+} = (B-A\eta)/(\ee^{\lambda_{+}}-\eta)\). Since \(\ee^{\lambda_{+}} = \ee^{-1}\ee^{r} = \ee^{-1}\bigl(1+O(\varepsilon) \bigr)\) by Equation (9.104) and \(\eta\) is negligible beside it, \(C_{+} = B\ee + O(\varepsilon)\) and hence \(C_{-} = A - B\ee + O(\varepsilon)\), every implied constant depending only on \(\abs{A}\), \(\abs{B}\) and \(c_{0}\).

Now compare term by term. For the outer term, on \(0\le x\le1\),

using \(\abs{\ee^{rx}-1}\le\abs{r}x\ee^{\abs{r}}\) and Equation (9.104). For the layer term, writing \(\ee^{\lambda_{-}x} = \ee^{-x/\varepsilon}\ee^{sx}\) and using \(\ee^{\lambda_{-}x}\le1\) for \(x\ge0\),

The first term is \(O(\varepsilon)\). In the second, \(\abs{\ee^{sx}-1}\le\abs{s}x\ee^{\abs{s}}\le2\ee^{2}x\) on \([0,1]\), and \(x\,\ee^{-x/\varepsilon}\) attains its maximum \(\varepsilon/\ee\) at \(x=\varepsilon\); so the second term is \(O(\varepsilon)\) too. Adding the two comparisons gives Equation (9.103).

Derivation. Derives Proposition 9.94. Use the expanded form of Remark 9.92 and substitute Equation (9.127) term by term, which is legitimate inside the disc of convergence (Theorem 7.51). Write \(J_{\nu} = \sum_{j}c_{j}x^{2j+\nu}\) with \(c_{j} = (-1)^{j}\bigl[j!\,\Gamma(j+\nu+1)\,2^{2j+\nu}\bigr]^{-1}\). The operator \(x^{2}\dd^{2}/\dd x^{2} + x\,\dd/\dd x - \nu^{2}\) sends \(x^{2j+\nu}\) to \(\bigl[(2j+\nu)^{2}-\nu^{2}\bigr]x^{2j+\nu} = 4j(j+\nu)\,x^{2j+\nu}\), while the remaining term \(x^{2}J_{\nu}\) shifts the index by one. The coefficient of \(x^{2j+\nu}\) in the whole expression is therefore \(4j(j+\nu)\,c_{j} + c_{j-1}\), and this vanishes: by \(j!\,\Gamma(j+\nu+1) = j\,(j-1)!\,(j+\nu)\,\Gamma(j+\nu)\) and \(2^{2j+\nu} = 4\cdot 2^{2(j-1)+\nu}\),

The term \(j = 0\) has no partner and vanishes on its own, because \(4j(j+\nu) = 0\) there.

Derivation. Derives Proposition 9.95. Both are term-by-term computations on Equation (9.127). For Equation (9.128),

whose derivative is \(\sum_{j\ge1}(-1)^{j}\,2j\,x^{2j-1} \bigl[j!\,\Gamma(j+\nu+1)2^{2j+\nu}\bigr]^{-1}\); putting \(j = i+1\) and cancelling the factor \(2(i+1)\) against \((i+1)!\) turns this into \(-\sum_{i\ge0}(-1)^{i}x^{2i+1} \bigl[i!\,\Gamma(i+\nu+2)\,2^{2i+\nu+1}\bigr]^{-1}\), which is \(-x^{-\nu}J_{\nu+1}(x)\) by Equation (9.127) at order \(\nu+1\). For Equation (9.129), the derivative of \(x^{\nu}J_{\nu} = \sum_{j}(-1)^{j}x^{2j+2\nu} \bigl[j!\,\Gamma(j+\nu+1)2^{2j+\nu}\bigr]^{-1}\) carries the factor \(2j+2\nu = 2(j+\nu)\), which cancels against \(\Gamma(j+\nu+1) = (j+\nu)\Gamma(j+\nu)\) and lowers the power of two by one, giving \(\sum_{j}(-1)^{j}x^{2j+2\nu-1} \bigl[j!\,\Gamma(j+\nu)2^{2j+\nu-1}\bigr]^{-1} = x^{\nu}J_{\nu-1}(x)\).

Derivation. Derives Corollary 9.96. Carrying out the differentiations on the left of Equations (9.128) and (9.129) and dividing by \(x^{-\nu}\) and \(x^{\nu}\) respectively gives

Subtracting gives Equation (9.130) and adding gives Equation (9.131). At a zero of \(J_{\nu}\) the first of the two displayed identities reduces to \(\dd J_{\nu}/\dd x = -J_{\nu+1}\).

Derivation. Derives Theorem 9.98. Take \(u(x) = J_{\nu}(\lambda x)\) and \(v(x) = J_{\nu}(\mu x)\) with \(\lambda,\mu > 0\), both solutions of Equation (9.132) with \(p(x) = x\), weight \(x\) and eigenvalues \(\lambda^{2}\) and \(\mu^{2}\). Integrating the Lagrange identity Equation (9.56) over \([0,a]\) gives

the prime denoting the derivative of \(J_{\nu}\) with respect to its own argument. The contribution at \(x = 0\) vanishes because \(p(x) = x\) does while \(J_{\nu}\) behaves there as \(x^{\nu}\) by Equation (9.127) — the second mechanism of Remark 9.58. Taking \(\lambda = \alpha_{\nu m}/a\) and \(\mu = \alpha_{\nu n}/a\) with \(m \neq n\) makes both Bessel values on the right vanish while the factor on the left does not, which is the off-diagonal statement; it is Theorem 9.57 with the boundary condition \(y(a)=0\), the first mechanism of Remark 9.58.

For the diagonal, fix \(\lambda = \alpha_{\nu m}/a\), keep \(\mu\) free and use \(J_{\nu}(\lambda a) = 0\) in Equation (9.134):

Let \(\mu \to \lambda\). The numerator vanishes like \(J_{\nu}(\mu a) = (\mu-\lambda)\,a\,J_{\nu}'(\lambda a) + \ldots\) and the denominator like \(-(\mu-\lambda)\,2\lambda\), so the quotient tends to \(\tfrac12 a^{2}\bigl[J_{\nu}'(\lambda a)\bigr]^{2}\). Finally \(J_{\nu}'(\alpha_{\nu m}) = -J_{\nu+1}(\alpha_{\nu m})\) by Corollary 9.96, and squaring removes the sign.

Derivation. Derives Proposition 9.99. Both factors of \(g(x,t) = \ee^{xt/2}\,\ee^{-x/(2t)}\) are entire in their arguments, so the two exponential series may be multiplied out and the resulting double series rearranged absolutely (Proposition 7.49):

the outer index being \(n = j-k\) and the inner sum running over those \(k \ge 0\) with \(k+n \ge 0\). Comparing with Equation (9.127) at integer order, where \(\Gamma(k+n+1) = (k+n)!\) and the reciprocal factorial is zero whenever \(k+n<0\), the inner sum is exactly \(J_{n}(x)\), which is Equation (9.135).

Derivation. Derives Proposition 9.100. Multiply Equation (9.136) by \(J_{\nu}(\alpha_{\nu n}x/a)\,x\) and integrate over \([0,a]\). Mean convergence permits the exchange of sum and integral, and Equation (9.133) leaves the single term \(m = n\) carrying the constant \(a^{2}\left[J_{\nu+1}(\alpha_{\nu n})\right]^{2}/2\). This is the argument of Proposition 9.48 run for a set that is orthogonal but not normalised, the normalising constant being divided out at the end instead of absorbed into the functions.

Derivation. Derives Proposition 9.102. Use the expanded form \(x^{2}y''+xy'+(x^{2}-\nu^{2})y = 0\) of Remark 9.92. With \(y = x^{-1/2}u\),

so \(x^{2}y'' = \tfrac34x^{-1/2}u - x^{1/2}u' + x^{3/2}u''\) and \(xy' = -\tfrac12x^{-1/2}u + x^{1/2}u'\). The terms in \(u'\) cancel, and adding \((x^{2}-\nu^{2})x^{-1/2}u\) leaves

which is Equation (9.138) after division by \(x^{3/2}\), since \(\tfrac34-\tfrac12 = \tfrac14\).

Derivation. Derives Proposition 9.103. Simplicity. On \(x>0\) the Bessel equation is regular — its leading coefficient \(x^{2}\) does not vanish — so Corollary 9.9 applies: a solution with \(y(x_{\ast}) = y'(x_{\ast}) = 0\) at some \(x_{\ast}>0\) is identically zero. Since \(J_{\nu}\) is not, no positive zero of it can be a zero of its derivative. (Equivalently, by Corollary 9.96, \(J_{\nu}' = -J_{\nu+1}\) at a zero of \(J_{\nu}\), and the two functions have no common positive zero.)

Location. Proposition 9.102 writes \(J_{\nu}(x) = x^{-1/2}u(x)\) with \(u\) solving an equation that tends to \(u''+u=0\): the solutions oscillate with asymptotic half-period \(\pi\), and the classical large-argument form quoted in Section 9.6.5 from [Morse:1953],

makes the phase explicit. The cosine vanishes where \(x - \nu\pi/2 - \pi/4 = \pi/2 + (n-1)\pi\), that is at \(x = (n+\nu/2-\tfrac14)\pi\), and it crosses each of those points with slope of modulus \(1\). A perturbation of size \(O(1/x)\) of a function crossing zero with unit slope displaces the crossing by \(O(1/x)\), and the amplitude \(\sqrt{2/\pi x}\), being non-zero, cannot create or destroy a crossing; so for large \(n\) there is exactly one zero of \(J_{\nu}\) within \(O(1/n)\) of each of those points, which is Equation (9.139), and there are infinitely many of them.

Derivation. Derives Proposition 9.106. Convergence is the ratio test (Proposition 7.48) exactly as for Equation (9.127), the moduli of the terms being the same. For the equation, run the computation of Proposition 9.94 with \(c_{j} = \bigl[j!\,\Gamma(j+\nu+1)2^{2j+\nu}\bigr]^{-1}\) and with the sign of the term \(x^{2}y\) reversed: the coefficient of \(x^{2j+\nu}\) becomes \(4j(j+\nu)c_{j} - c_{j-1}\), and the identity \(j!\,\Gamma(j+\nu+1) = j\,(j-1)!\,(j+\nu)\,\Gamma(j+\nu)\) used there gives \(4j(j+\nu)c_{j} = c_{j-1}\), so it vanishes; the term \(j=0\) vanishes on its own.

Every coefficient in Equation (9.142) is strictly positive for \(\nu\ge0\), and so is every term for \(x>0\); the same is true of the term-by-term derivative, legitimate inside the radius of convergence (Theorem 7.51). Hence \(I_{\nu}>0\) and \(I_{\nu}'>0\) on \(x>0\), which forbids a zero there. The leading power is \((x/2)^{\nu}/\Gamma(\nu+1)\), giving the stated behaviour at the origin. Finally \(I_{\nu}\) and \(K_{\nu}\) are independent — one is bounded at infinity and the other is not, by Remark 9.107 — so by Theorem 9.13 they span the two-dimensional solution space.

Derivation. Derives Proposition 9.109. Write \(v = \left(x^{2}-1\right)^{n}\), a polynomial of degree \(2n\) whose \(n\)-th derivative has degree \(n\); the coefficient of \(x^{n}\) comes from \(\dd^{n}x^{2n}/\dd x^{n} = \bigl[(2n)!/n!\bigr]x^{n}\), which after the division by \(2^{n}n!\) gives the stated leading coefficient. For the value at \(1\), factor \(v = (x-1)^{n}(x+1)^{n}\): in the Leibniz expansion of \(\dd^{n}v/\dd x^{n}\) every term retains at least one factor \(x-1\) except the one in which all \(n\) derivatives fall on \((x-1)^{n}\), so at \(x = 1\) only \(n!\,(x+1)^{n} = n!\,2^{n}\) survives, and division by \(2^{n}n!\) gives \(P_{n}(1) = 1\).

For the equation, start from the identity \(\left(x^{2}-1\right)\dv{v}{x} = 2n\,x\,v\), which is immediate from the definition of \(v\), and differentiate it \(n+1\) times by the Leibniz rule Equation (7.16). On the left only three terms survive, because \(x^{2}-1\) is quadratic:

and on the right only two, because \(x\) is linear: \(2n\,x\,v^{(n+1)} + 2n(n+1)\,v^{(n)}\). Writing \(u = v^{(n)}\) and collecting,

which on multiplication by \(-1\) is \(\bigl[(1-x^{2})u'\bigr]' + n(n+1)u = 0\), that is Equation (9.146). Since \(P_{n}\) is a constant multiple of \(u\), it solves the same linear equation.

Derivation. Derives Lemma 9.110. Let \(\Lambda[y] = \bigl[(1-x^{2})y'\bigr]'\). On monomials

so \(\Lambda\) maps the space of polynomials of degree at most \(N\) into itself and is triangular in the monomial basis ordered by degree, with diagonal entries \(-k(k+1)\) for \(k = 0,\ldots,N\). Those entries are pairwise distinct, so each eigenvalue of \(\Lambda\) on that space is simple and its eigenspace is one-dimensional. A polynomial solution of Equation (9.146) of degree at most \(N\) is an eigenvector for the eigenvalue \(-n(n+1)\), and \(P_{n}\) is one such by Proposition 9.109; taking \(N\) at least as large as the degree of the given solution identifies the two up to a constant.

Derivation. Derives Theorem 9.111. For the off-diagonal terms read Equation (9.146) as the eigenvalue problem it is: \(p(x) = 1-x^{2}\), \(q = 0\), weight \(r = 1\) and eigenvalue \(\lambda^{2} = n(n+1)\), distinct for distinct non-negative degrees. The coefficient function \(p\) vanishes at both endpoints while \(P_{m}\), \(P_{n}\) and their derivatives are polynomials and hence bounded there, so Equation (9.57) holds by the second mechanism of Remark 9.58 and Theorem 9.57 gives the vanishing of the integral for \(m \neq n\).

For the diagonal use Equation (9.148) and integrate by parts \(n\) times (Equation (7.28)). At each step the integrated term carries a factor \(\dd^{j}(x^{2}-1)^{n}/\dd x^{j}\) with \(j < n\), which retains a factor \(\left(x^{2}-1\right)\) and so vanishes at \(x = \pm1\). Hence

since the \(2n\)-th derivative of a monic-leading polynomial of degree \(2n\) is the constant \((2n)!\) and \((-1)^{n}(x^{2}-1)^{n} = (1-x^{2})^{n}\). Write \(I_{n} = \int_{-1}^{1}(1-x^{2})^{n}\,\dd x\). Integration by parts gives

the boundary term vanishing and \(x^{2} = 1 - (1-x^{2})\) splitting the integral; hence \(I_{n} = \bigl[2n/(2n+1)\bigr]I_{n-1}\), and with \(I_{0} = 2\),

the closed form following from \(\prod_{k\le n}2k = 2^{n}n!\) and \(\prod_{k\le n}(2k+1) = (2n+1)!/\left(2^{n}n!\right)\). Multiplying,

Derivation. Derives Theorem 9.113. Write \(W = 1-2xt+t^{2}\), so that \(g = W^{-1/2}\). For \(\abs{x}\le1\) the roots of \(W\) in \(t\) are \(\ee^{\pm\ii\theta}\) with \(x = \cos\theta\), both of modulus one, so \(g\) is holomorphic in \(t\) on \(\abs{t}<1\) and has there a Taylor expansion \(g = \sum_{n\ge0}p_{n}(x)\,t^{n}\) (Theorem 8.20). Expanding \(\bigl[1 - t(2x-t)\bigr]^{-1/2}\) by the binomial series and collecting powers of \(t\) shows each \(p_{n}\) to be a polynomial in \(x\) of degree at most \(n\).

Next, \(g\) satisfies the identity

Indeed \(\pp g/\pp x = t\,W^{-3/2}\) and \(\pp g/\pp t = (x-t)\,W^{-3/2}\), so the first bracket differentiates to \(-2xt\,W^{-3/2} + 3\left(1-x^{2}\right)t^{2}W^{-5/2}\) and the second to \(2t(x-t)W^{-3/2} - t^{2}W^{-3/2} + 3t^{2}(x-t)^{2}W^{-5/2}\). The terms in \(W^{-3/2}\) add to \(-3t^{2}W^{-3/2}\), and those in \(W^{-5/2}\) add to \(3t^{2}\bigl[(1-x^{2}) + (x-t)^{2}\bigr]W^{-5/2} = 3t^{2}W\cdot W^{-5/2}\), which cancels the first.

Substituting the Taylor series in that identity and equating coefficients of \(t^{n}\), using \(\pp_{t}\bigl[t^{2}\pp_{t}g\bigr] = \sum_{n}n(n+1)p_{n}t^{n}\), gives

which is Equation (9.146). So \(p_{n}\) is a polynomial solution of the Legendre equation of that degree, hence a constant multiple of \(P_{n}\) by Lemma 9.110. The constant is \(1\) because \(g(1,t) = (1-t)^{-1} = \sum_{n}t^{n}\) gives \(p_{n}(1) = 1\), and \(P_{n}(1) = 1\) by Proposition 9.109.

Derivation. Derives Proposition 9.114. Differentiating Equation (9.150) with respect to \(t\) gives \(W\,\pp g/\pp t = (x-t)\,g\), and with respect to \(x\) gives \(W\,\pp g/\pp x = t\,g\). Substituting the series and reading off the coefficient of \(t^{n}\) turns the first into

which is Equation (9.151), and the second into

Differentiating Equation (9.151) gives \((n+1)P_{n+1}' = (2n+1)P_{n} + (2n+1)xP_{n}' - nP_{n-1}'\), and Equation (9.153) at index \(n+1\) gives \(2xP_{n}' = P_{n+1}' + P_{n-1}' - P_{n}\). Eliminating \(xP_{n}'\) between the two leaves \(\tfrac12 P_{n+1}' = \tfrac12(2n+1)P_{n} + \tfrac12 P_{n-1}'\), which is Equation (9.152). The initial values follow from \(g(x,t) = 1 + xt + O\left(t^{2}\right)\).

Derivation. Derives Proposition 9.115. Multiply Equation (9.154) by \(P_{m}\) and integrate over \([-1,1]\); mean convergence permits the exchange of sum and integral, and Equation (9.149) leaves the single term \(n = m\) with the factor \(2/(2m+1)\), which inverts to Equation (9.155). That an expansion exists for every square-integrable \(f\) — the completeness of the system — is again a Sturm–Liouville theorem and not proved here, exactly as in Remark 9.101.

Derivation. Derives Lemma 9.117. Differentiate \(\left(1-x^{2}\right)y'' - 2xy' + n(n+1)y = 0\) exactly \(m\) times by the Leibniz rule Equation (7.16). The first term contributes \(\left(1-x^{2}\right)y^{(m+2)} - 2mx\,y^{(m+1)} - m(m-1)y^{(m)}\), the second \(-2x\,y^{(m+1)} - 2m\,y^{(m)}\), and the third \(n(n+1)y^{(m)}\); only three and two terms survive respectively because \(1-x^{2}\) is quadratic and \(x\) linear. Collecting, the coefficient of \(y^{(m+1)}\) is \(-2(m+1)x\) and that of \(y^{(m)}\) is \(n(n+1) - m(m-1) - 2m = n(n+1) - m(m+1)\).

Derivation. Derives Proposition 9.118. Put \(\varpi = \left(1-x^{2}\right)^{m/2}\) and \(y = \varpi\,u\) with \(u = \dd^{m}P_{n}/\dd x^{m}\), so that \(\varpi'/\varpi = -mx/\left(1-x^{2}\right)\) and

Substituting \(y = \varpi u\) into the associated equation Equation (9.157), written out as \(\left(1-x^{2}\right)y'' - 2xy' + \bigl[n(n+1) - m^{2}/(1-x^{2})\bigr]y = 0\), and dividing by \(\varpi\) leaves an equation for \(u\) whose coefficient of \(\dd u/\dd x\) is \(2\left(1-x^{2}\right)\varpi'/\varpi - 2x = -2(m+1)x\) and whose coefficient of \(u\) is

that is \(n(n+1) - m(m+1)\). The result is exactly Equation (9.160), which \(u\) satisfies by Lemma 9.117.

Derivation. Derives Theorem 9.119. For \(n \neq n'\) read Equation (9.157) at fixed \(m\) as the eigenvalue problem with \(p(x) = 1-x^{2}\), \(q(x) = -m^{2}/\left(1-x^{2}\right)\), weight \(r = 1\) and eigenvalue \(\lambda^{2} = n(n+1)\). The functions \(P_{n}^{m}\) carry the factor \(\left(1-x^{2}\right)^{m/2}\), so for \(m \ge 1\) both they and \(p\) times their derivatives vanish at \(x = \pm1\), while for \(m = 0\) the argument is that of Theorem 9.111; either way Equation (9.57) holds and Theorem 9.57 applies.

For the diagonal put \(I_{m} = \int_{-1}^{1}\left(1-x^{2}\right)^{m} \bigl[u_{m}(x)\bigr]^{2}\,\dd x\) with \(u_{m} = \dd^{m}P_{n}/\dd x^{m}\), which is the integral to be computed because \(\left(P_{n}^{m}\right)^{2} = \left(1-x^{2}\right)^{m}u_{m}^{2}\). For \(m \ge 1\) write \(u_{m} = \dd u_{m-1}/\dd x\) and integrate by parts (Equation (7.28)); the boundary term carries the factor \(\left(1-x^{2}\right)^{m}\) and vanishes, so

where \(\Theta = \left(1-x^{2}\right)\dd^{2}u_{m-1}/\dd x^{2} - 2m\,x\,\dd u_{m-1}/\dd x\). By Equation (9.160) at order \(m-1\) that combination is \(-\bigl[n(n+1) - m(m-1)\bigr]u_{m-1}\), so

Iterating from \(I_{0} = 2/(2n+1)\), which is Theorem 9.111, gives

which is Equation (9.161).

Derivation. Derives Proposition 9.120. Write \(D = \dd/\dd x\). Applying \(D^{m}\) to Equation (9.151) by the Leibniz rule Equation (7.16), in which the factor \(x\) contributes only two terms,

while applying \(D^{m-1}\) to Equation (9.152) gives \(D^{m}P_{n+1} - D^{m}P_{n-1} = (2n+1)\,D^{m-1}P_{n}\). Substituting the second into the first eliminates \(D^{m-1}P_{n}\) and leaves

and multiplying through by \(\left(1-x^{2}\right)^{m/2}\) turns each \(D^{m}P_{k}\) into \(P_{k}^{m}\) by Equation (9.159).

Derivation. Derives Proposition 9.121. The angular eigenvalue equation is

Substituting the product and dividing by \(\Theta\Phi/\sin^{2}\theta\) separates it: the \(\varphi\) part is \(\dd^{2}\Phi/\dd\varphi^{2} = -m^{2}\Phi\), whose solutions are the exponentials Equation (9.107), and single-valuedness under \(\varphi \mapsto \varphi + 2\pi\) restricts \(m\) to the integers. The \(\theta\) part, in the variable \(x = \cos\theta\) — for which \(\dd/\dd\theta = -\sin\theta\,\dd/\dd x\) and \(\sin^{2}\theta = 1-x^{2}\) — is exactly the associated Legendre equation Equation (9.157) with \(n(n+1)\) replaced by \(\Lambda\). Boundedness at the two poles \(x = \pm1\), the singular endpoints of that equation in the sense of Remark 9.17, selects the solutions \(P_{n}^{m}\): for \(\Lambda = n(n+1)\) with \(n\) a non-negative integer and \(\abs{m}\le n\) these are bounded by Equation (9.159), and the second solution \(Q_{n}^{m}\) diverges there, as does every solution for any other value of \(\Lambda\).

Derivation. Derives Theorem 9.122. The integral factorises. The azimuthal factor is

which is Equation (9.109) on an interval of one period, so every term with \(m \neq m'\) vanishes whatever the polar factor does. At \(m = m'\) the polar factor is, substituting \(x = \cos\theta\) and \(\dd x = -\sin\theta\,\dd\theta\),

by Theorem 9.119. The product of the two factors is \(2\pi \cdot 2(n+m)!\bigl/\bigl[(2n+1)(n-m)!\bigr]\) times \(\delta_{nn'}\delta_{mm'}\), and the squared modulus of the constant in Equation (9.163) is the reciprocal of that number, which gives Equation (9.164). Since the constant enters squared and is required positive, it is unique.

Derivation. Derives Proposition 9.125. Complete the square: \(2xt-t^{2} = x^{2} - (x-t)^{2}\), so the left of Equation (9.169) is \(\ee^{x^{2}}\,\ee^{-(x-t)^{2}}\). The second factor is a function of \(x-t\) alone, so \(\pp/\pp t\) acting on it equals \(-\pp/\pp x\); hence its \(n\)-th \(t\)-derivative at \(t = 0\) is \((-1)^{n}\dd^{n}\ee^{-x^{2}}/\dd x^{n}\), and the Taylor expansion in \(t\) of a function entire in \(t\) (Theorem 8.20) gives

which is Equation (9.169) with Equation (9.168). Differentiating Equation (9.168) once more, and using \(\dd\ee^{-x^{2}}/\dd x = -2x\,\ee^{-x^{2}}\), gives \(H_{n+1} = 2x\,H_{n} - \dd H_{n}/\dd x\); with \(H_{0} = 1\) this shows by induction that \(H_{n}\) is a polynomial of degree \(n\) whose leading coefficient doubles at each step, hence equals \(2^{n}\). The parity follows from \(\ee^{2(-x)t-t^{2}} = \ee^{2x(-t)-(-t)^{2}}\), which sends \(H_{n}(-x)\) to \((-1)^{n}H_{n}(x)\).

Derivation. Derives Proposition 9.126. Differentiating Equation (9.169) with respect to \(x\) gives \(2t\,\ee^{2xt-t^{2}} = \sum_{n}\bigl(\dd H_{n}/\dd x\bigr)t^{n}/n!\), and comparing coefficients of \(t^{n}\) gives \(\dd H_{n}/\dd x\bigm/n! = 2H_{n-1}/(n-1)!\), which is Equation (9.170). Combining it with \(H_{n+1} = 2xH_{n} - \dd H_{n}/\dd x\), established in the proof of Proposition 9.125, gives Equation (9.171). For the equation, differentiate that same relation: \(\dd H_{n+1}/\dd x = 2H_{n} + 2x\,\dd H_{n}/\dd x - \dd^{2}H_{n}/\dd x^{2}\), and use Equation (9.170) on the left, which turns it into \(2(n+1)H_{n}\). Cancelling \(2H_{n}\) from both sides leaves \(\dd^{2}H_{n}/\dd x^{2} - 2x\,\dd H_{n}/\dd x + 2n\,H_{n} = 0\).

Derivation. Derives Theorem 9.127. For \(m \neq n\) apply Theorem 9.57 with the data Equation (9.167): the eigenvalues \(2m\) and \(2n\) differ, and the boundary term Equation (9.57) vanishes because \(p = \ee^{-x^{2}}\) multiplies polynomials and \(\ee^{-x^{2}}\) beats every polynomial at both ends of the line — the third mechanism of Remark 9.58.

For \(m = n\) insert Equation (9.168) for one factor and integrate by parts \(n\) times (Equation (7.28)), every boundary term vanishing for the same reason:

By Proposition 9.125 the \(n\)-th derivative of \(H_{n}\) is the constant \(2^{n}n!\), and the Gaussian integral \(\int_{-\infty}^{\infty}\ee^{-x^{2}}\dd x = \sqrt{\pi}\) — its square being \(\int\int\ee^{-x^{2}-y^{2}}\dd x\,\dd y = \pi\) in plane polar coordinates — completes Equation (9.172).

Derivation. Derives Proposition 9.128. Multiply by \(H_{m}(x)\ee^{-x^{2}}\), integrate over the line, exchange sum and integral by mean convergence, and use Equation (9.172); the surviving term is \(n = m\) and carries the constant \(2^{m}m!\sqrt{\pi}\). Completeness of the system carries the same caveat as Remark 9.101.

Derivation. Derives Proposition 9.131. By the Leibniz rule Equation (7.16),

and dividing by \(n!\,\ee^{-x}\) and putting \(j = n-k\) gives Equation (9.177); the term \(j=n\) gives the leading coefficient and the term \(j=0\) gives \(L_{n}(0) = 1\). For the generating function expand the exponential and then each factor \((1-t)^{-1-j}\) by the binomial series,

both series being absolutely convergent for \(\abs{t}<1\), so that they may be rearranged (Proposition 7.49). The coefficient of \(t^{n}\) collects \(i = n-j\) and is \(\sum_{j\le n}\binom{n}{j}(-x)^{j}/j!\), which is Equation (9.177).

Derivation. Derives Proposition 9.132. Write \(v = x^{n}\ee^{-x}\), which satisfies \(x\,\dd v/\dd x = (n-x)\,v\). Differentiating that identity \(n+1\) times by the Leibniz rule Equation (7.16), each side producing only two terms because \(x\) and \(n-x\) are linear, gives for \(u = v^{(n)}\)

that is \(x\,u'' + (1+x)\,u' + (n+1)\,u = 0\). Now \(u = n!\,\ee^{-x}L_{n}\) by Equation (9.176), so \(u' = n!\,\ee^{-x}\left(L_{n}' - L_{n}\right)\) and \(u'' = n!\,\ee^{-x}\left(L_{n}'' - 2L_{n}' + L_{n}\right)\); substituting and cancelling \(n!\ee^{-x}\),

which is Equation (9.174). For the recurrence, write \(G\) for the left of Equation (9.178); then \(\ln G = -\ln(1-t) - xt/(1-t)\) gives \(\pp G/\pp t = G\bigl[(1-t)^{-1} - x(1-t)^{-2}\bigr]\), that is \(\left(1-t\right)^{2}\pp G/\pp t = \left(1-t-x\right)G\). Substituting the series and reading off the coefficient of \(t^{n}\) gives \((n+1)L_{n+1} - 2n\,L_{n} + (n-1)L_{n-1} = L_{n} - L_{n-1} - x\,L_{n}\), which rearranges to Equation (9.179).

Derivation. Derives Theorem 9.133. For \(m \neq n\) apply Theorem 9.57 with the data Equation (9.175): the eigenvalues \(m\) and \(n\) differ, and \(p(x) = x\ee^{-x}\) vanishes at the origin and decays at infinity while the solutions are polynomials, so Equation (9.57) holds.

For \(m = n\) use Equation (9.176) on one factor and integrate by parts \(n\) times (Equation (7.28)). Every boundary term vanishes: at infinity because of \(\ee^{-x}\), and at the origin because each derivative \(\dd^{k}\bigl(x^{n}\ee^{-x}\bigr)/\dd x^{k}\) with \(k<n\) still carries a positive power of \(x\). Hence

By Equation (9.177) the \(n\)-th derivative of \(L_{n}\) is the constant \((-1)^{n}\), and \(\int_{0}^{\infty}x^{n}\ee^{-x}\dd x = \Gamma(n+1) = n!\), so the whole expression is \(\left[(-1)^{n}\right]^{2}n!/n! = 1\).

Derivation. Derives Proposition 9.134. This is Proposition 9.48 verbatim: the system \(\set{L_{n}}\) is already orthonormal for the weight \(\ee^{-x}\) by Equation (9.180), so no normalising constant appears. Completeness carries the caveat of Remark 9.101.

Derivation. Derives Lemma 9.136. Write \(W_{n}\) for the integral. Integrating by parts (Corollary 7.44) with \(u = \sin^{2n-1}\theta\) and \(\dd v = \sin\theta\,\dd\theta\), the boundary term \(\bigl[-\sin^{2n-1}\theta\cos\theta\bigr]_{0}^{\pi/2}\) vanishes at both ends for \(n\ge1\), leaving

by \(\cos^{2} = 1 - \sin^{2}\) (Lemma 7.71). Hence \(2n\,W_{n} = (2n-1)W_{n-1}\), and with \(W_{0} = \pi/2\) induction gives

the last step because \(\prod_{j\le n}(2j-1) = (2n)!/\bigl(2^{n}n!\bigr)\) and \(\prod_{j\le n}(2j) = 2^{n}n!\).

Derivation. Derives Proposition 9.139. The binomial series \((1-u)^{-1/2} = \sum_{n\ge0}\bigl[(2n)!/(4^{n}(n!)^{2})\bigr]u^{n}\) converges for \(\abs{u}<1\) (Theorem 7.50, Proposition 7.48: the ratio of successive coefficients is \((2n+1)/(2n+2)\to1\)). Put \(u = k^{2}\sin^{2}\theta\), so that \(0\le u\le k^{2}<1\) uniformly in \(\theta\). The \(n\)-th term is bounded on \([0,\pi/2]\) by \(M_{n} = \bigl[(2n)!/(4^{n}(n!)^{2})\bigr]k^{2n}\), and \(\sum_{n}M_{n} = (1-k^{2})^{-1/2}\) is finite, so by Lemma 9.6 the series converges uniformly on \([0,\pi/2]\) and may be integrated term by term. Each term contributes \(\bigl[(2n)!/(4^{n}(n!)^{2})\bigr]k^{2n}W_{n}\) with \(W_{n}\) the Wallis integral Equation (9.181), and substituting Equation (9.181) produces the square in Equation (9.186). The first coefficients are \(1\), \((1/2)^{2}=1/4\), \((3/8)^{2}=9/64\) and \((5/16)^{2}=25/256\).

Monotonicity is immediate: the integrand of Equation (9.182) increases with \(k\) at every \(\theta\in(0,\pi/2]\). For the divergence, use \(1-k^{2}\sin^{2}\theta = (1-k^{2}) + k^{2}\cos^{2}\theta \le c^{2} + \cos^{2}\theta\) with \(c = \sqrt{1-k^{2}}\), and substitute \(w = \pi/2-\theta\), so \(\cos\theta = \sin w \le w\):

which tends to \(+\infty\) as \(c\to0^{+}\), that is as \(k\to1^{-}\).

Derivation. Derives Proposition 9.141. The first identity in Equation (9.188) is the Pythagorean identity (Lemma 7.71) applied to \(\operatorname{am}\), and the second is Equation (9.187) squared. Since \(\pp F/\pp\varphi = (1-k^{2}\sin^{2}\varphi)^{-1/2}\) is positive and continuous, Proposition 7.32 gives

and the chain rule (Proposition 7.31) turns Equation (9.187) into Equation (9.189). Differentiating \(\operatorname{dn}^{2} = 1-k^{2}\operatorname{sn}^{2}\) and dividing by \(2\operatorname{dn}\) gives the third derivative, \(-k^{2}\operatorname{sn}\operatorname{cn}\).

\(F\) is odd in \(\varphi\) because its integrand is even, so \(\operatorname{am}\) is odd and the parities follow. The values at \(u=0\) are those at \(\varphi=0\), and \(\operatorname{am}(K) = \pi/2\) by Equation (9.185), giving \(\operatorname{sn}K = 1\). Finally \(F(\varphi+\pi) = F(\varphi)+2K\) inverts to \(\operatorname{am}(u+2K) = \operatorname{am}(u)+\pi\), and Equation (9.190) is what \(\sin\), \(\cos\) and \(\sin^{2}\) do under a shift by \(\pi\). At \(k=0\), Equation (9.182) gives \(F(\varphi,0)=\varphi\) and hence \(\operatorname{am}(u,0)=u\).

Derivation. Derives Proposition 9.142. Squaring Equation (9.189) and eliminating \(\operatorname{cn}\) and \(\operatorname{dn}\) by Equation (9.188) gives Equation (9.192). Differentiating \(\operatorname{sn}' = \operatorname{cn}\operatorname{dn}\) once more and using the three derivative formulas,

which is Equation (9.193). The initial values are those of Proposition 9.141, with \(\operatorname{sn}'(0) = \operatorname{cn}0\,\operatorname{dn}0 = 1\). The right-hand side of Equation (9.193) is a polynomial and therefore locally Lipschitz, so Theorem 9.8 makes the solution unique. Note that Equation (9.192), whose right-hand side has a square root at the turning points, does not by itself determine the solution there; the second-order form does.

Derivation. Derives Proposition 9.143. Substitute \(v = e_{1} + (e_{2}-e_{1})\sin^{2}\theta\) (Corollary 7.44), which increases from \(e_{1}\) to \(e_{2}\) as \(\theta\) runs over \([0,\pi/2]\). Then

and, using Equation (9.194), \(v-e_{3} = -(e_{3}-e_{1})\bigl[1-k^{2}\sin^{2}\theta\bigr]\). Hence

whose square root is \(2\omega(e_{2}-e_{1})\sin\theta\cos\theta \sqrt{1-k^{2}\sin^{2}\theta}\) on \([0,\pi/2]\), every factor being non-negative there. Since \(\dd v = 2(e_{2}-e_{1})\sin\theta\cos\theta\,\dd\theta\), everything cancels except

and integrating from \(0\) to \(\varphi\) gives Equation (9.195), finite at both ends because the integrand of Equation (9.182) is bounded for \(k<1\). Taking \(\varphi=\pi/2\) gives \(K(k)/\omega\).

Inverting Equation (9.195) means \(\varphi = \operatorname{am}(\omega t,k)\), hence \(\sin^{2}\varphi = \operatorname{sn}^{2}(\omega t,k)\), which is Equation (9.196). Differentiating it with Equation (9.189), \(\dd u/\dd t = 2\omega(e_{2}-e_{1}) \operatorname{sn}\operatorname{cn}\operatorname{dn}\), whose square is \(4\omega^{2}(e_{2}-e_{1})^{2} \operatorname{sn}^{2}\operatorname{cn}^{2}\operatorname{dn}^{2}\); and the same substitutions that produced \(f(v)\) above, now with \(\sin\theta = \operatorname{sn}\), give \(f(u) = c(e_{2}-e_{1})^{2}(e_{3}-e_{1}) \operatorname{sn}^{2}\operatorname{cn}^{2}\operatorname{dn}^{2}\). The two agree precisely when \(4\omega^{2} = c(e_{3}-e_{1})\), which is Equation (9.194). Finally \(\operatorname{sn}^{2}\) has period \(2K\) by Equation (9.190), so \(u\) has period \(2K/\omega\).

Derivation. Derives Proposition 9.144. Multiplying Equation (9.197) by \(\dd\theta/\dd t\) and integrating once gives the first integral \(\tfrac12\dot\theta^{2} - \omega_{0}^{2}\cos\theta = \text{const}\). The stated initial data fix the constant so that \(\dot\theta\) vanishes at \(\theta = \pm\theta_{0}\), and with \(\cos\theta = 1-2\sin^{2}(\theta/2)\),

The right-hand side is negative beyond \(\abs{\theta} = \theta_{0}\), so the motion is confined there and reverses at the two turning points. Equation (9.197) is unchanged under \(t\mapsto-t\), and also under \(\theta\mapsto-\theta\) because the sine is odd, so the four quarters of the motion are congruent and the period is four times the time from \(\theta=0\) to \(\theta=\theta_{0}\).

On that quarter \(\dot\theta>0\). Substitute \(\sin(\theta/2) = k\sin\varphi\), which carries \(\theta: 0\to\theta_{0}\) into \(\varphi: 0\to\pi/2\) and gives \(\tfrac12\cos(\theta/2)\,\dd\theta = k\cos\varphi\,\dd\varphi\) with \(\cos(\theta/2) = \sqrt{1-k^{2}\sin^{2}\varphi}\), while Equation (9.201) gives \(\dot\theta = 2\omega_{0}k\cos\varphi\). Hence

so \(\omega_{0}t = F(\varphi,k)\), that is \(\varphi = \operatorname{am}(\omega_{0}t,k)\), which is Equation (9.198); and \(\varphi=\pi/2\) at the turning point gives \(T/4 = K(k)/\omega_{0}\), which is Equation (9.199).

For Equation (9.200), expand \(k = \sin(\theta_{0}/2)\) by Taylor's theorem (Theorem 7.38): \(k^{2} = \theta_{0}^{2}/4 - \theta_{0}^{4}/48 + O(\theta_{0}^{6})\) and \(k^{4} = \theta_{0}^{4}/16 + O(\theta_{0}^{6})\). Substituting into Equation (9.186) truncated at \(k^{4}\),

and \(-1/192 + 9/1024 = (-16+27)/3072 = 11/3072\).