Question

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.

Answer #1

Prove that a function f(z) which is complex differentiable at a
point z0 satisfies the Cauchy-Riemann equations at that point.

Two functions, u(x,y) and v(x,y), are said to verify the
Cauchy-Riemann
differentiation equations if they satisfy the following
equations ∂u\dx=∂v/dy and ∂u/dy=−(∂v/dx)
a. Verify that the Cauchy-Riemann differentiation equations can
be written in the polar coordinate form as
∂u/dr=1/dr ∂v/dθ and ∂v/dr =−1/r ∂u/∂θ
b. Show that the following functions satisfy the Cauchy-Riemann
differen- tiation equations
u=ln sqrt(x^(2)+y^(2)) and v= arctan y/x.

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.

Complex Analysis Proof - Prove the uniqueness of the limit.

Can you find a functionv (x,y) so that u+iv is entire with
u(x,y) =x^3+ 3xy^2 ? (cauchy- riemann equations---hint)

(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)|.

Complex Analysis Proof - Prove: A set is closed if and only if S
contains all of its accumulation points.

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?

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

Verify the Caucy-riemann equations for the functions u(x,y),
v(x,y) defined in the given domain
u(x,y)=x³-3xy², v(x,y)=3x²y-y³, (x,y)ɛR
u(x,y)=sinxcosy,v(x,y)=cosxsiny (x,y)ɛR
u(x,y)=x/(x²+y²), v(x,y)=-y/(x²+y²),(x²+y²), (
x²+y²)≠0
u(x,y)=1/2 log(x²+y²), v(x,y)=sin¯¹(y/√¯x²+y²), ( x˃0 )
In each case,state a complex functions whose real and imaginary
parts are u(x,y) and v(x,y)

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