Question

a) If F(x) is an analytic function and either Re(F(z)) or Im(F(z)) is a constant, then F (z) is a constant function.

Answer #1

13. Show that an analytic function f(z) in a domain D cannot
have a constant modulus unless f is a constant function.

For
function, f(z)=ze^z
use Cauchy-Reimann equations to see if its analytic and find
the derivative of f(z)

Please show all steps, thank you:
Problem C: Does there exist an analytic function f(z) in some
domain D with the real part u(x,y)=x^2+y^2?
Problem D: Is the function f(z)=(x-iy)^2 analytic in any domain
in C? Are the real part u(x,y) and the imaginary pary v(x,y)
harmonic in C? Are u and v harmonic conjugates of each other in any
domain?

Consider a function F=u+iv which is analytic on the set
D={z|Rez>1} and that u_x+v_y=0 on D. Show that there exists a
real constant p and a complex constant q such that F(z)=-ipz+q on
D.
Notation: Here u_x denotes the partial derivative of u with
respect to x and v_y denotes the partial derivative of v with
respect to y.

Are the following function harmonic? If your answer is yes, find
a corresponding analytic function f (z) =u(x, y) + iv(x, y). v = (
2x + 1)y

Find a function f(x,y,z) such that ∇f is the constant vector
〈8,3,5〉.

Find the set of complex numbers z satisfying the two conditions:
Re((z+1)^2)=0 and Im((z−1)^2)=2. Here Re(a + bi) = a and Im(a + bi)
= b if both a, b ∈ R. Then find the cardinality of the set.

Find v(x,y) so that f(z) = 3x^2 +8xy - 3y^2 + iv(x,y) is
analytic

Assume set A={z1,z2,....,zm) is a m-point set in C. If f(z) is
analytic and bounded on C\A, prove that f(z) always equal to a
constant

(1) Find all functions f(z) that are analytic in the entire
complex plane and satisfy 2|sin(z)| ≥ |f(z)|.
(2) Find all functions f(z) that are analytic in the entire
complex plane and satisfy 2|f(z)| ≥ |sin(z)|.

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