12. Use differentiation formulas from calculus to find f'(z). a) f(2) = 322 – 2z+ 4i b) f(2) = (2z + 5)(z+i)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Question 12
Use the definition f'(z0) = lim (2) - f (Zo) of the derivative to find the
Find the image of the semi-infinite strip x20, 0s ysA under the
6.
transformation w= e², and label the corresponding portions of the boundaries.
7.
Use the ɛ,8 - definition of a limit to prove the following limit.
lim (az +b) = az, +b, for complex numbers a and b with a + 0.
8.
Use Limit Laws to evaluate the following limits:
iz -1
422
a) lim
b)
lim (z? - 4z +2+ 5i)
c)
lim
2+0 (z–1)?
z+2+i
4z° +z
lim
9.
Prove the following limit.
= 00
2+ 00 z+i
10.
Use the precise ɛ,8 - definition of continuity to prove that if f(z) is continuous
at zo then f(z) is continuous at zo -
11.
z- z0
derivative of f(z) =
for z +0.
12.
Use differentiation formulas from calculus to find f'(z).
f(2) = 3z2 – 2z +4i
a)
b)
f(z) = (2z+5)(z+i)³
az +b
13.
Find the derivative of the function T(z) =
where a, b, c, d are complex
cz +d'
numbers such that ad – bc + 0. When is T'(z) = 0?
14.
Use the Cauchy-Riemann Equations to show that f'(z) does not exist at any point.
a)
f(z) = Z
b)
f(2) = e*eiy
15.
Use the Cauchy-Riemann Equations to show that f(z) = z Im(z) is only
differentiable at z = 0 and find the value of f'(0).
16.
Use the Cauchy-Riemann Equations to show that f(z) = z is differentiable for
all z and find f'(z).
Transcribed Image Text:Use the definition f'(z0) = lim (2) - f (Zo) of the derivative to find the Find the image of the semi-infinite strip x20, 0s ysA under the 6. transformation w= e², and label the corresponding portions of the boundaries. 7. Use the ɛ,8 - definition of a limit to prove the following limit. lim (az +b) = az, +b, for complex numbers a and b with a + 0. 8. Use Limit Laws to evaluate the following limits: iz -1 422 a) lim b) lim (z? - 4z +2+ 5i) c) lim 2+0 (z–1)? z+2+i 4z° +z lim 9. Prove the following limit. = 00 2+ 00 z+i 10. Use the precise ɛ,8 - definition of continuity to prove that if f(z) is continuous at zo then f(z) is continuous at zo - 11. z- z0 derivative of f(z) = for z +0. 12. Use differentiation formulas from calculus to find f'(z). f(2) = 3z2 – 2z +4i a) b) f(z) = (2z+5)(z+i)³ az +b 13. Find the derivative of the function T(z) = where a, b, c, d are complex cz +d' numbers such that ad – bc + 0. When is T'(z) = 0? 14. Use the Cauchy-Riemann Equations to show that f'(z) does not exist at any point. a) f(z) = Z b) f(2) = e*eiy 15. Use the Cauchy-Riemann Equations to show that f(z) = z Im(z) is only differentiable at z = 0 and find the value of f'(0). 16. Use the Cauchy-Riemann Equations to show that f(z) = z is differentiable for all z and find f'(z).
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