Show the following functions are holomorphic on the domains given, and compute their derivative f'(z): (a) U = C, f(z) = (1+2z+32²)7. (b) U = C\ {2i}, f(z) = 21 (c) U = C\ {i, -i}, ƒ(z) = (¹–2²¹. (d) U = {z € C: cos(z) 0} and f(z) =tan(z) where tan(z) is defined to be (e) U = C and f(z) = cosh(z) where cosh(z) is defined as cos(iz). sin(z) COR(2)
Show the following functions are holomorphic on the domains given, and compute their derivative f'(z): (a) U = C, f(z) = (1+2z+32²)7. (b) U = C\ {2i}, f(z) = 21 (c) U = C\ {i, -i}, ƒ(z) = (¹–2²¹. (d) U = {z € C: cos(z) 0} and f(z) =tan(z) where tan(z) is defined to be (e) U = C and f(z) = cosh(z) where cosh(z) is defined as cos(iz). sin(z) COR(2)
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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q1
![1.
Show the following functions are holomorphic on the domains given, and compute their derivatives
f'(z):
(a) U = C, f(z) = (1+2z+32²)7.
(b) U = C\ {2i), ƒ(z) = 22-1
(c) U = C\ {i, -i}, f(x) = (4=22².
(d) U = {z EC: cos(z) 0} and f(z) =tan(z) where tan(z) is defined to be sin(2)
COS(2)
(e) U = C and f(z) = cosh(z) where cosh(z) is defined as cos(iz).
[Hint: You should be able to do most of these using Proposition 2.7.7.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27adff13-7954-4bd1-a3e1-459b28324214%2F9e13b7b0-c345-4b3c-a11e-936af09cd016%2Fkx4z7q6_processed.png&w=3840&q=75)
Transcribed Image Text:1.
Show the following functions are holomorphic on the domains given, and compute their derivatives
f'(z):
(a) U = C, f(z) = (1+2z+32²)7.
(b) U = C\ {2i), ƒ(z) = 22-1
(c) U = C\ {i, -i}, f(x) = (4=22².
(d) U = {z EC: cos(z) 0} and f(z) =tan(z) where tan(z) is defined to be sin(2)
COS(2)
(e) U = C and f(z) = cosh(z) where cosh(z) is defined as cos(iz).
[Hint: You should be able to do most of these using Proposition 2.7.7.]
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