(a) (b) Prove that for every complex number u E C, we have un 2 (47)² = 27₁/²+1 dz for n = 0, 1,2,3,..., uneuz n! 2πί n!zn+1 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 8 2 1 2π Σ (17) ² = 2²/2 √²h 2²u cos0 do. e n! 2π JO n=0 (Hint: Consider the function eu(z+1/2). Expand the function eu/z in a power series with respect to u and then apply part (a).)
(a) (b) Prove that for every complex number u E C, we have un 2 (47)² = 27₁/²+1 dz for n = 0, 1,2,3,..., uneuz n! 2πί n!zn+1 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 8 2 1 2π Σ (17) ² = 2²/2 √²h 2²u cos0 do. e n! 2π JO n=0 (Hint: Consider the function eu(z+1/2). Expand the function eu/z in a power series with respect to u and then apply part (a).)
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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Question
help with part b please. thank you
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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