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Chapter 7: Q10P (page 384)

Find the exponential Fourier transform of the given f(x)and write f(x)as a Fourier integral.

Short Answer

Expert verified

Answer

The exponential Fourier transform of the given function is gα2iπα2αa-sinαaand fxas a Fourier integral isfx=∫-∞∞αa-sinαaiπα2eiαxdα.

Step by step solution

01

Given information

The graph of the given function is as shown below.

In mathematical form, the function can be written as shown below.

fx=-2x+a,x∈-a,0-2x-a,x∈0,a.

The exponential Fourier transform of the function is to be found and the function is to be written as a Fourier integral.

02

The significance of Fourier transforms

The following are the formulas for the Fourier series transforms.

Here,is called the Fourier transform of.

03

Find the Fourier transform

Use the formula gα=22π∫-∞∞fxe-iαxdxto find the Fourier transform.

Solve further to obtain.

Now, writeas a Fourier integral using the formula.

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Most popular questions from this chapter

The diagram shows a “relaxation” oscillator. The chargeqon the capacitor builds up until the neon tube fires and discharges the capacitor (we assume instantaneously). Then the cycle repeats itself over and over.

(a) The charge q on the capacitor satisfies the differential equation

, here R is the Resistance, C is the capacitance and Vis the

Constant d-c voltage, as shown in the diagram. Show that if q=0 when

t=0 then at any later time t (during one cycle, that is, before the neon

Tube fires),

(b) Suppose the neon tube fires at. Sketch q as a function of t for

several cycles.

(b) Expand the periodic q in part (b) in an appropriate Fourier series.

For each of the periodic functions in Problems5.1to 5.11, use Dirichlet's theorem to find the value to which the Fourier series converges at x=0,±π/2,±π,±2π.

In Problems 17to 20,find the Fourier sine transform of the function in the indicated problem, and write f(x)as a Fourier integral [use equation (12.14)]. Verify that the sine integral for f(x)is the same as the exponential integral found previously.

Problem 10.

In each of the following problems you are given a function on the interval -π<x<π .Sketch several periods of the corresponding periodic function of period 2ττ . Expand the periodic function in a sine-cosine Fourier series,

f(x)={-x,-π<x<0x,0<x<π

In each case, show that a particle whose coordinate is (a) x=Re(z) , (b)y=lmzis undergoing simple harmonic motion, and find the amplitude, period, frequency, and velocity amplitude of the motion.

role="math" localid="1659242473978" -4ei(2t+3π)
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