Question

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

Answer #1

1. Let D ⊂ C be an open set and let γ be a circle contained in
D. Suppose f is holomorphic on D except possibly at a point z0
inside γ. Prove that if f is bounded near z0, then
f(z)dz = 0. γ
2. The function f(z) = e1/z has an essential singularity at z =
0. Verify the truth of Picard’s great theorem for f. In other
words, show that for any w ∈ C (with possibly...

Complex Variable: Schwarz's Theorem
Show that if f(z) is analytic for ∣z∣≤R, f(0)=0 and M ∣f(z)∣≤M
then ∣f(z)∣≤ ((M lz∣ )/R).
(detailed please)

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.

Prove:
f(x) has a minimum value of m on the set A ⊂ Rn at the point x =
z if and only if −f(x) has a maximum value of −m on A at the same
point x = 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?

Find the maximum and minimum values of f(x,y,z)=2x-2y-z on the
closed and bounded set 4x^2+2y^2+z^2 ≤ 1

Let f(z) and g(z) be entire functions, with |f(z) - g(z)| < M
for some positive real number M and all z in C. Prove that f'(z) =
g'(z) for all z in C.

Let F = {A ⊆ Z : |A| < ∞} be the set of all finite sets of
integers. Let R be the relation on F defined by A R B if and only
if |A| = |B|. (a) Prove or disprove: R is reflexive. (b) Prove or
disprove: R is irreflexive. (c) Prove or disprove: R is symmetric.
(d) Prove or disprove: R is antisymmetric. (e) Prove or disprove: R
is transitive. (f) Is R an equivalence relation? Is...

for the surface
f(x/y/z)=x3+3x2y2+y3+4xy-z2=0
find any vector that is normal to the surface at the point
Q(1,1,3). use this to find the equation of the tangent plane to the
surface at q.

problem 2 In the polynomial ring Z[x], let I = {a0 + a1x + ... +
anx^n: ai in Z[x],a0 = 5n}, that is, the set of all polynomials
where the constant coefficient is a multiple of 5. You can assume
that I is an ideal of Z[x]. a. What is the simplest form of an
element in the quotient ring z[x] / I? b. Explicitly give the
elements in Z[x] / I. c. Prove that I is not a...

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