f(z) - f(z,) Use the definition f'(zo) = lim of the derivative to find the 02 - 2 derivative of f(z) =- for z +0.

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 11 Queation 12
Use the definition f'(zo) = lim (2)- f(2o) of the derivative to find the
6.
Find the image of the semi-infinite strip x20, 0s ysA under the
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
lim (z? -4z +2+ 5i)
a) lim
2i z+i
b)
c)
lim
2+0 (z–1)?
z+2+i
4z° +z
lim
9.
Prove the following limit.
= 00
z+ 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.
where a, b, c, d are complex
Find the derivative of the function T(z) =
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.
а)
f(z) = 7
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'(zo) = lim (2)- f(2o) of the derivative to find the 6. Find the image of the semi-infinite strip x20, 0s ysA under the 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 lim (z? -4z +2+ 5i) a) lim 2i z+i b) c) lim 2+0 (z–1)? z+2+i 4z° +z lim 9. Prove the following limit. = 00 z+ 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. where a, b, c, d are complex Find the derivative of the function T(z) = 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. а) f(z) = 7 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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