6.1. (a) Write the function f(z) = 2³ +1 in the form f(z) = u(x, y) +iv(x, y). (b) Write the function f(z) = x + 2iy2 as a function involving z and z. 1 (c) Find f(1 - 2i) if f(2)= + Z. 22-1
6.1. (a) Write the function f(z) = 2³ +1 in the form f(z) = u(x, y) +iv(x, y). (b) Write the function f(z) = x + 2iy2 as a function involving z and z. 1 (c) Find f(1 - 2i) if f(2)= + Z. 22-1
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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Transcribed Image Text:Exercises
6.1. (a) Write the function f(z) = z³+1 in the form f(z) = u(x, y) +iv(x, y).
(b) Write the function f(z) = x + 2iy2 as a function involving z and z.
1
(c) Find f(1-2i) if f(x) =
+2.
2² - 1
6.2. Determine where the following functions are analytic and find their
derivatives:
(a) f(z) = (2+1)4¹ + 1/z.
1
(z + 2i) ² (z² + 1)
(b) f(2)=
Transcribed Image Text:68 6 Analytic Functions
(c) f(2)= 10(2iz² + 5) 6.
(d) f(2)= x³ - 3xy² +1+i(3x²y - y³).
(e) f(z) = |2|².
6.3. If fu+iv is non-constant and analytic in an open set U, which one
of the following are analytic as well?
(a) g = u - iv,
(b) h=v+iu,
(c) k=-u- iv,
(d) l= iu - v.
6.4. Using the Cauchy-Riemann criterion show that f(2)= 2³ is entire.
6.5. Let f(z) = (x³ + 3x²y + y³) +i(-x³-3xy²).
(a) Use Cauchy-Riemann equations to find all points at hich
tiable. Show these points graphically.
(b) Is there any open set in which f is differentiable? If yes, find one such
open set. Is f analytic anywhere?
6.6. For each case find the region in which f is analytic and find f'.
(a) f(2)=
zn/n!.
n=0²
(b) f(2)=non!z".
(-1)n-1
(c) f(z) =
n
1
(d) f(2)=n=0 2n (z − 1)n
1
(e) f(2)= x=0
n+1
n+i 2n
6.7. Suppose that the radius of convergence of the power series
o anz
is equal to 1. If we know that f(k) (0) = 0 for all k ≥ 1, and f(0) = 1, find
f(1/2+1).
2n=0
2n
6.8. Show that f(2)= Eno(z+1)/n³ is analytic in D₁(-1) and find f'(2).
6.9. Find where the function
f(2)=z-
is analytic. Show that f'(z) =
6.10. Show that f(z) =
equation
23 25 27
3 5
1
z²+1
22n
(n!)²
2² f"(z) + z f'(z) = 42² f(z).
differen-
2n=0
is entire and satisfies the differential
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