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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