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(a) Prove that for every complex number u € C, we have 2 1 (43)² = 2 + + Sc Son! un euz dz n! 2πί n!zn+1 where C is a closed path which contains the origin in its interior. for n = 0, 1, 2, 3, ...,

(a) Prove that for every complex number u € C, we have 2 1 (43)² = 2 + + Sc Son! un euz dz n! 2πί n!zn+1 where C is a closed path which contains the origin in its interior. for n = 0, 1, 2, 3, ...,

Advanced Engineering Mathematics
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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(a) (b) Prove that for every complex number u € C, we have 2 un 1 (²7) ² - 2 + √² = n! 2πί uneuz dz for n = 0, 1, 2, 3,..., where C is a closed path which contains the origin in its interior. (Hint: Use the Cauchy Integral Formula for the nth derivative.) Prove that it holds 1 •2πT 2 ()² = ²/6² Σ n! 2πT n=0 e2u cos de. (Hint: Consider the function eu(z+¹/z). Expand the function eu/z in a power series with respect to u and then apply part (a).)
Transcribed Image Text:(a) (b) Prove that for every complex number u € C, we have 2 un 1 (²7) ² - 2 + √² = n! 2πί uneuz dz for n = 0, 1, 2, 3,..., where C is a closed path which contains the origin in its interior. (Hint: Use the Cauchy Integral Formula for the nth derivative.) Prove that it holds 1 •2πT 2 ()² = ²/6² Σ n! 2πT n=0 e2u cos de. (Hint: Consider the function eu(z+¹/z). Expand the function eu/z in a power series with respect to u and then apply part (a).)
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