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Chapter 1: Q45MP (page 1)

Prove the following shifting or translation theorems for Fourier transforms. If g(伪) is the Fourier transform of f(x), then

  1. The Fourier transform of f(x-a)is e-颈伪补g(伪);

  2. The Fourier transform of e颈尾虫f(x)is g(伪-尾).

Short Answer

Expert verified
  1. The Fourier transform of fx-ais e-i伪ag伪;
  2. The Fourier transform of ei尾xfxis g伪-尾.

Step by step solution

01

Given Information

Fourier transform is given by,

ft=鈭�-鈭�鈭�fte-jwtdt=鈭�-鈭�鈭�fte-iwtdt

Calculate the Fourier transform offx+a=鈭�-鈭�鈭�fx+ae-iwxdx

02

Fourier Transformation

A Fourier transform (FT) is a mathematical transformation that decomposes functions that are spatially or temporally dependent into functions that are spatially or temporally dependent.

03

Fourier Transformation for a

a)

Let the equation x+a=t

dx=dt

=鈭�-鈭�鈭�fte-iwt-伪dt=鈭�-鈭�鈭�fte-iat-adt=eia伪鈭�-鈭�鈭�fte-iatdtg伪=鈭�-鈭�鈭�fte-iatdt=eiaag伪

04

Fourier Transformation for b

b)

The Fourier transform is given by

ei尾xfx=鈭�-鈭�鈭�ei尾xfxe-iaxdx=鈭�-鈭�鈭�fxe-ia+尾xdx=g伪+尾

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Most popular questions from this chapter

Test the following series for convergence

鈭�n-1鈭�(-1)nnn+5

Use the preliminary test to decide whether the following series are divergent or require further testing. Careful:Do notsay that a series is convergent; the preliminary test cannot decide this.

6.鈭�n=1鈭�n!(n+1)!

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

d10dx10(x8tan2x)atx=0.

By computer or tables, find the exact sum of each of the following series.

(a)鈭�n=1鈭�n(4n2-1)2(b)鈭�n=1鈭�n3n!(c)鈭�n=1鈭�n(n+1)3n

Test the following series for convergence.

3.鈭�n=1鈭�(鈭�1)nn2
See all solutions

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