Mathematical Methods for Physics and Engineering : A Comprehensive Guide

12.10 HINTS AND ANSWERS (a) (b) (c) (d) 00110 1240 Figure 12.6 Continuations of exp(−x^2 )in0≤x≤1 to give: (a) cosine terms only ...

FOURIER SERIES 12.21 cn=[(−1)nsinhπ]/[π(1 +n^2 )].Having setx= 0, separate out then=0term and note that (−1)n=(−1)−n. 12.23 (π^2 ...

13 Integral transforms In the previous chapter we encountered the Fourier series representation of a periodic function in a fixe ...

INTEGRAL TRANSFORMS c(ω)expiωt − 1 0 0 1 2 r −^2 Tπ^2 Tπ^4 Tπ ωr Figure 13.1 The relationship between the Fourier terms for a fu ...

13.1 FOURIER TRANSFORMS and (13.3) becomes f(t)= 1 2 π ∫∞ −∞ dω eiωt ∫∞ −∞ du f(u)e−iωu. (13.4) This result is known asFourier’s ...

INTEGRAL TRANSFORMS
Find the Fourier transform of the normalised Gaussian distribution f(t)= 1 τ √ 2 π exp ( − t^2 2 τ^2 ) , −∞ ...

13.1 FOURIER TRANSFORMS is a wavefunction and the distribution of the wave intensity in time is given by |f|^2 (also a Gaussian) ...

INTEGRAL TRANSFORMS −Y Y y k x k′ 0 θ Figure 13.2 Diffraction grating of width 2Ywith light of wavelength 2π/k being diffracted ...

13.1 FOURIER TRANSFORMS f(y) 1 −a−b −a+b a−b a a+b −a x Figure 13.3 The aperture functionf(y) for two wide slits. After some man ...

INTEGRAL TRANSFORMS equals zero. This leads immediately to two further useful results: ∫b −a δ(t)dt= 1 for alla, b > 0 (13.13 ...

13.1 FOURIER TRANSFORMS The derivative of the delta function,δ′(t), is defined by ∫∞ −∞ f(t)δ′(t)dt= [ f(t)δ(t) ]∞ −∞ − ∫∞ −∞ f′ ...

INTEGRAL TRANSFORMS
Prove relation (13.23). Considering the integral ∫∞ −∞ f(t)H′(t)dt= [ f(t)H(t) ]∞ −∞ − ∫∞ −∞ f′(t)H(t)dt =f ...

13.1 FOURIER TRANSFORMS ω (a) (b) −Ω Ω π t Ω 1 ̃fΩ fΩ(t) 2Ω (2π)^1 /^2 Figure 13.4 (a) A Fourier transform showing a rectangular ...

INTEGRAL TRANSFORMS (i) Differentiation: F [ f′(t) ] =iω ̃f(ω). (13.28) This may be extended to higher derivatives, so that F [ ...

13.1 FOURIER TRANSFORMS Ignoring in the present context the effect of the termAaexp(iωct), which gives a contribution to the tra ...

INTEGRAL TRANSFORMS g(y) (a) (b) (c) (d) 0 y Figure 13.5 Resolution functions: (a) idealδ-function; (b) typical unbiased resolut ...

13.1 FOURIER TRANSFORMS −a aa−a x y z f(x) −b b 2 b 2 b ∗ g(y) = h(z) 1 Figure 13.6 The convolution of two functionsf(x)andg(y). ...

INTEGRAL TRANSFORMS given by ̃h(k)=√^1 2 π ∫∞ −∞ dz e−ikz {∫∞ −∞ f(x)g(z−x)dx } = 1 √ 2 π ∫∞ −∞ dx f(x) {∫∞ −∞ g(z−x)e−ikzdz } . ...

13.1 FOURIER TRANSFORMS The inverse of convolution, called deconvolution, allows us to find a true distributionf(x) given an obs ...

INTEGRAL TRANSFORMS
Prove the Wiener–Kinchin theorem, ̃C(k)= √ 2 π[ ̃f(k)]∗ ̃g(k). (13.42) Following a method similar to that f ...