proposition 17.61 Mean-square response of a damped resonator

Rests on

• depends_on proposition 17.52 Linear translation-invariant systems ¶
  • depends_on definition 17.50 Convolution ¶
  • depends_on definition 17.38 Schwartz space and tempered distributions ¶
  • depends_on proposition 17.41 The identities physics uses ¶
    • depends_on definition 17.40 Transform and derivative of a distribution ¶
      • depends_on definition 17.38 Schwartz space and tempered distributions ¶ ↺
      • depends_on proposition 17.39 The transform preserves $\mathcal{S}$ ¶
        • depends_on definition 17.38 Schwartz space and tempered distributions ¶ ↺
        • depends_on proposition 17.30 Elementary properties ¶
        • depends_on theorem 17.32 Fourier inversion ¶
        • proves proof ch:15-fourier-integral-transforms@proof-25 ¶
    • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      • depends_on definition 7.41 Antiderivative ¶
        • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
      • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
        • depends_on theorem 7.40 Continuous functions are integrable ¶
        • depends_on theorem 7.24 Extreme value theorem ¶
        • proves proof ch:05-real-analysis@proof-24 ¶
      • proves proof ch:05-real-analysis@proof-25 ¶
    • depends_on theorem 17.32 Fourier inversion ¶ ↺
    • proves proof ch:15-fourier-integral-transforms@proof-26 ¶
  • proves proof ch:15-fourier-integral-transforms@proof-33 ¶
• depends_on theorem 8.24 Residue theorem ¶
  • depends_on corollary 8.15 Deformation of contours ¶
    • depends_on definition 6.17 Simply connected space ¶
      • depends_on definition 6.6 Continuous map ¶
        • depends_on definition 3.53 Preimage ¶
        • depends_on definition 6.2 Open set ¶
        • depends_on definition 6.1 Topological space ¶
      • depends_on definition 6.15 Path-connected space ¶
        • depends_on definition 6.6 Continuous map ¶ ↺
    • depends_on theorem 8.12 Cauchy ¶
      • depends_on equation 8.5 eq:cpx-cr ¶
      • depends_on theorem 7.131 Green ¶
        • depends_on definition 7.127 Simple regions ¶
        • depends_on remark 7.128 What the derivations below take as given ¶
        • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶ ↺
        • proves proof ch:05-real-analysis@proof-77 ¶
      • proves proof ch:06-complex-analysis@proof-7 ¶
    • proves proof ch:06-complex-analysis@proof-9 ¶
  • depends_on definition 8.22 Isolated singularities; residue ¶
    • depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
      • depends_on equation 8.6 eq:cpx-contour-integral ¶
      • depends_on proposition 8.4 Euler's formula ¶
        • depends_on equation 8.2 eq:cpx-exp-series ¶
        • depends_on proposition 7.47 Comparison; absolute convergence ¶
        • proves proof ch:06-complex-analysis@proof-2 ¶
      • proves proof ch:06-complex-analysis@proof-6 ¶
    • depends_on theorem 8.21 Laurent expansion ¶
      • depends_on corollary 8.15 Deformation of contours ¶ ↺
      • depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
      • depends_on theorem 8.16 Cauchy integral formula ¶
        • depends_on corollary 8.15 Deformation of contours ¶ ↺
        • depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
        • depends_on lemma 8.9 ML estimate ¶
        • proves proof ch:06-complex-analysis@proof-10 ¶
      • depends_on theorem 8.20 Taylor expansion ¶
        • depends_on equation 8.12 eq:cpx-cif ¶
        • depends_on equation 8.13 eq:cpx-derivative-formula ¶
        • depends_on lemma 8.9 ML estimate ¶ ↺
        • depends_on proposition 7.46 Geometric series ¶
        • proves proof ch:06-complex-analysis@proof-14 ¶
      • proves proof ch:06-complex-analysis@proof-15 ¶
  • depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
  • depends_on theorem 8.21 Laurent expansion ¶ ↺
  • proves proof ch:06-complex-analysis@proof-17 ¶
• depends_on theorem 17.59 Wiener–Khinchin ¶
  • depends_on definition 17.58 Stationary process and power spectral density ¶
    • depends_on definition 17.56 Correlation ¶
    • depends_on definition 17.29 Fourier transform; the treatise convention ¶
  • depends_on definition 17.29 Fourier transform; the treatise convention ¶ ↺
  • depends_on theorem 17.32 Fourier inversion ¶ ↺
  • proves proof ch:15-fourier-integral-transforms@proof-36 ¶
• proves proof ch:15-fourier-integral-transforms@proof-37 ¶

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