Consider the (real-valued) function f(x):= on the interval - ≤ x ≤, where a > 1 is real. a. Extend f(x) to a function f(z) on the complex plane. Find and classify all singular points z EC of this extended function. That is, where are the singular points, and what types of singularities (simple poles, double poles, essential singularities, etc.) are they? b. Recall that the Fourier series coefficients for f(x) are defined by the integrals f(x) e-in² da. În == cos x a - sin x 1 2/7 2π Show that fn = 0, and that fin are complex conjugates of one another. c. Relate the Fourier coefficients fn for n>0 to the closed contour integrals f(z)e-inz dz, Jn (R) := # $cm)² 2π where C(R) denotes the closed, rectangular contour in the complex plane with its corners at z = ±ñ and z = ± - iR. Why do we choose to close the rectangle in the lower half-plane? d. Evaluate fn by applying the residue theorem of Jn (R).
Consider the (real-valued) function f(x):= on the interval - ≤ x ≤, where a > 1 is real. a. Extend f(x) to a function f(z) on the complex plane. Find and classify all singular points z EC of this extended function. That is, where are the singular points, and what types of singularities (simple poles, double poles, essential singularities, etc.) are they? b. Recall that the Fourier series coefficients for f(x) are defined by the integrals f(x) e-in² da. În == cos x a - sin x 1 2/7 2π Show that fn = 0, and that fin are complex conjugates of one another. c. Relate the Fourier coefficients fn for n>0 to the closed contour integrals f(z)e-inz dz, Jn (R) := # $cm)² 2π where C(R) denotes the closed, rectangular contour in the complex plane with its corners at z = ±ñ and z = ± - iR. Why do we choose to close the rectangle in the lower half-plane? d. Evaluate fn by applying the residue theorem of Jn (R).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please write clearly and explain all the steps! Thank you! Please part c and d.
The course is about complex variables.
please show boundaries and in the complex plane system
Transcribed Image Text:Consider the (real-valued) function
COs x
f(x) :=
а — sin a
on the interval –a <x< T, where a > 1 is real.
a. Extend f(x) to a function f(2) on the complex plane. Find and classify all singular points
zeC of this extended function. That is, where are the singular points, and what types of
singularities (simple poles, double poles, essential singularities, etc.) are they?
b. Recall that the Fourier series coefficients for f(x) are defined by the integrals
1
În :=
f(x) e¯inz
dr.
Show that fn = 0, and that f±n are complex conjugates of one another.
c. Relate the Fourier coefficients f, for n > 0 to the closed contour integrals
1
:= f(2) e¬in= dz,
(R)
where C(R) denotes the closed, rectangular contour in the complex plane with its corners at
z = ±n and z = ±n – iR. Why do we choose to close the rectangle in the lower half-plane?
d. Evaluate fn by applying the residue theorem of Jn(R).
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