Question

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

Answer #1

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.

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.

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

Complex Analysis Proof - Prove: if f = u + iv is analytic in a
domain D, then u and v satisfy the Cauchy-Riemann equations in
D.

Let f be a function with measurable domain D. Then f is
measurable if and only if the function g(x)={f(x) if x\in D ,0 if x
\notin D } is measurable.

Suppose z is holomorphic in a bounded domain, continuous on the
closure, and constant on the boundary. Prove that f must be
constant throughout the domain.

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

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

I have a function. The function domain is all finite subsets of
integer Z. The codomain natrual number (0, 1,2, ...n). the function
itself is h(S) = cardinality of S. This function is apparently not
surjective nor injective. How can I change the domain so that this
cardinality function is both surjective and injective? I want to
keep the domain as large as possible.
Thanks

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