Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.1 FOURIER TRANSFORMS obtained simply by noting from (13.42) that the cross-correlation (13.40) of two functionsfandgcanbewrit ...

INTEGRAL TRANSFORMS two- or three-dimensional functions of position. For example, in three dimensions we can define the Fourier ...

13.2 LAPLACE TRANSFORMS A similar result may be obtained for two-dimensional Fourier transforms in whichf(r)=f(ρ), i.e.f(r) is i ...

INTEGRAL TRANSFORMS (iii) Once again using the definition (13.53) we have f ̄n(s)= ∫∞ 0 tne−stdt. Integrating by parts we find f ...

13.2 LAPLACE TRANSFORMS f(t) f ̄(s) s 0 cc/s 0 ctn cn!/sn+1 0 sinbt b/(s^2 +b^2 )0 cosbt s/(s^2 +b^2 )0 eat 1 /(s−a) a tneat n!/ ...

INTEGRAL TRANSFORMS
Find the Laplace transform ofd^2 f/dt^2. Using the definition of the Laplace transform and integrating by p ...

13.2 LAPLACE TRANSFORMS We may now consider the effect of multiplying the Laplace transformf ̄(s)by e−bs(b>0). From the defin ...

INTEGRAL TRANSFORMS Figure 13.7 Two representations of the Laplace transform convolution (see text). where the integral in the b ...

13.3 CONCLUDING REMARKS The properties of the Laplace transform derived in this section can sometimes be useful in finding the L ...

INTEGRAL TRANSFORMS 13.4 Exercises 13.1 Find the Fourier transform of the functionf(t)=exp(−|t|). (a) By applying Fourier’s inve ...

13.4 EXERCISES Determine the convolution offwith itself and, without further integration, deduce its transform. Deduce that ∫∞ − ...

INTEGRAL TRANSFORMS 13.10 In many applications in which the frequency spectrum of an analogue signal is required, the best that ...

13.4 EXERCISES (a) Find the Fourier transform of f(γ, p, t)= { e−γtsinpt t > 0 , 0 t< 0 , whereγ(>0) andpare constant p ...

INTEGRAL TRANSFORMS 13.18 The equivalent duration and bandwidth,TeandBe, of a signalx(t) are defined in terms of the latter and ...

13.4 EXERCISES (c) L[sinhatcosbt]=a(s^2 −a^2 +b^2 )[(s−a)^2 +b^2 ]−^1 [(s+a)^2 +b^2 ]−^1. 13.24 Find the solution (the so-called ...

INTEGRAL TRANSFORMS 13.27 The functionfa(x) is defined as unity for 0<x<aand zero otherwise. Find its Laplace transformf ̄ ...

13.5 HINTS AND ANSWERS 13.17 V ̃(k)∝[− 2 π/(ik)] ∫ {exp[−(μ−ik)r]−exp[−(μ+ik)r]}dr. 13.19 Note that the lower limit in the calcu ...

14 First-order ordinary differential equations Differential equations are the group of equations that contain derivatives. Chap- ...

14.1 GENERAL FORM OF SOLUTION the application of some suitableboundary conditions. For example, we may be told that for a certai ...

FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS In the case of (14.2), we have dy dx =a 1 cosx−a 2 sinx, d^2 y dx^2 =−a 1 sinx−a 2 c ...