equation 17.42 eq:ft-gaussian

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• depends_on lemma A.599 One variable, complex coefficient ¶
  • depends_on lemma A.601 Several variables, complex symmetric matrix ¶
    • depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
    • depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
      • depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
        • depends_on proposition A.609 The exponent is a quadratic polynomial ¶
• depends_on lemma A.584 Fourier injectivity in $n$ variables ¶
  • depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
    • depends_on lemma A.590 The Gram matrix is universal ¶
      • depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
    • depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      • depends_on lemma A.590 The Gram matrix is universal ¶ ↺
      • depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
    • depends_on proposition A.592 The vacuum of the Schrödinger system ¶
• depends_on lemma A.586 Gaussian integral with a complex linear term ¶
  • depends_on proposition A.592 The vacuum of the Schrödinger system ¶ ↺
  • depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶ ↺
• depends_on lemma 106.2 Gaussian integrals ¶
  • depends_on lemma 106.1 The short-time kernel ¶
    • depends_on theorem 106.3 Feynman's sum over histories ¶
      • depends_on phenomenon 106.5 The Aharonov–Bohm effect ¶
      • depends_on theorem 106.16 Feynman–Kac ¶
        • depends_on proposition 106.17 Euclidean time projects onto the ground state ¶
        • depends_on proposition 106.21 Reflection positivity is the positivity of the Hilbert space ¶
  • depends_on theorem 106.27 The free generating functional and the Feynman propagator ¶
  • depends_on theorem 106.54 The gauge-fixed action ¶
    • depends_on proposition 106.55 Ghosts are required by unitarity ¶
• depends_on lemma 100.28 The $d$-dimensional loop integral ¶
  • depends_on proposition 106.45 One-loop effective potential ¶
  • depends_on theorem 100.35 One-loop electron self-energy ¶
    • depends_on phenomenon 100.66 The Lamb shift ¶
  • depends_on theorem 100.31 One-loop vacuum polarization ¶
    • depends_on phenomenon 100.69 Light scatters off light ¶
• depends_on theorem 17.32 Fourier inversion ¶
  • depends_on corollary 17.81 Sampling periodises the spectrum; aliasing ¶
  • depends_on corollary 17.33 The transform is injective ¶
  • depends_on lemma A.584 Fourier injectivity in $n$ variables ¶ ↺
  • depends_on proposition 17.41 The identities physics uses ¶
    • depends_on proposition 17.90 Hankel transform as the axially symmetric Fourier transform ¶
    • depends_on proposition 17.52 Linear translation-invariant systems ¶
      • depends_on definition 28.25 Complex susceptibility ¶
      • depends_on proposition 17.61 Mean-square response of a damped resonator ¶
  • depends_on proposition 17.42 Sokhotski–Plemelj ¶
    • depends_on proposition 105.22 The two-state effective generator ¶
      • depends_on proposition 105.23 Symmetry constraints on the mass matrix ¶
    • depends_on theorem 17.78 Causality implies dispersion relations ¶
      • depends_on remark 33.10 Fibre anelasticity, and why it biases one method and not the other ¶
      • depends_on remark 28.26 Causality is visible in the susceptibility ¶
      • depends_on remark 28.62 Neither velocity is a speed limit ¶
  • depends_on proposition 17.39 The transform preserves $\mathcal{S}$ ¶
    • depends_on definition 17.40 Transform and derivative of a distribution ¶
      • depends_on definition 10.42 Green's function ¶
        • depends_on definition 10.50 Retarded and advanced Green's functions ¶
        • depends_on example 10.49 The method of images ¶
        • depends_on proposition 28.24 Impulse response of the damped oscillator ¶
        • depends_on proposition 10.47 Bilinear expansion ¶
        • depends_on proposition 10.43 Representation of the solution ¶
        • depends_on theorem 10.45 Symmetry of the Green's function ¶
      • depends_on proposition 17.41 The identities physics uses ¶ ↺
  • depends_on theorem 17.69 Bromwich inversion integral ¶
    • depends_on corollary 17.71 Residue evaluation and causality ¶
      • depends_on corollary 17.76 Heaviside's expansion theorem ¶
      • depends_on proposition 17.77 Poles and stability ¶
  • depends_on theorem 17.53 Convolution theorem ¶
    • depends_on theorem 17.57 Wiener–Khinchin, finite-energy form ¶
    • depends_on theorem 10.62 The heat kernel ¶
      • depends_on proposition 10.63 Infinite propagation speed ¶
  • depends_on theorem 17.106 Filtered back-projection in a plane slice ¶
    • depends_on proposition 25.59 Homodyne tomography inverts a Radon transform ¶
  • depends_on theorem 17.94 Mellin inversion ¶
    • depends_on proposition 17.96 Poles give asymptotics ¶
    • depends_on theorem 17.100 Moments diagonalise a convolution evolution ¶
      • depends_on corollary 17.101 Coupled channels ¶
  • depends_on theorem 17.36 Plancherel ¶
    • depends_on theorem 17.43 Bandwidth theorem ¶
      • depends_on corollary 17.44 Duration and bandwidth in SI ¶
      • depends_on remark 17.46 The quantum reading, in SI ¶
  • depends_on theorem 17.105 Radon inversion in three-dimensional space ¶
  • … 3 more
• depends_on theorem 17.43 Bandwidth theorem ¶ ↺
• depends_on theorem 10.62 The heat kernel ¶ ↺