Question

The real part of a f (z) complex function is given as
(x,y)=y^3-3x^2y. Show the harmonic function u(x,y) and find the
expressions v(x,y) and f(z). Calculate f'(1+2i) and write x+iy
algebraically.

Answer #1

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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?

Show that the function f(z) =x^3-3xy^2+i((3x^2y-y^3) is
differentiable

Using the derivative definition, point the derivative value for
the given function f(z)=3/z^2 Find in Z0=1+i and write x+iy
algebraically.

Suppose f is entire, with real and imaginary parts u and v
satisfying u(x, y) v(x, y) = 3 for
all z = x + iy. Show that f is constant.
Be clearly, please. Do not upload same answers from others on
Chegg. THANKS

Let X and Y have a joint density function given by f(x; y) = 3x;
0 <= y <= x <= 1
(a) Find P(X<2Y).
(b) Find cov(X,Y).
(c) Find P(X < 1/2 |Y = 1/3).
(d) Find P(X = 1/2|Y = 1/3).
(e) Find P(X > 1/2|Y > 1/3).
(f) Find the conditional expectation E(X|Y = y).

For the given function u(x, y) = cos(ax) sinh(3y),(a >
0);
(a) Find the value of a such that u(x, y) is harmonic.
(b) Find the harmonic conjugate of u(x, y) as v(x, y).
(c) Find the analytic function f(z) = u(x, y) + iv(x, y) in
terms of z.
(d) Find f ′′( π 4 − i) =?

part 1)
Find the partial derivatives of the function
f(x,y)=xsin(7x^6y):
fx(x,y)=
fy(x,y)=
part 2)
Find the partial derivatives of the function
f(x,y)=x^6y^6/x^2+y^2
fx(x,y)=
fy(x,y)=
part 3)
Find all first- and second-order partial derivatives of the
function f(x,y)=2x^2y^2−2x^2+5y
fx(x,y)=
fy(x,y)=
fxx(x,y)=
fxy(x,y)=
fyy(x,y)=
part 4)
Find all first- and second-order partial derivatives of the
function f(x,y)=9ye^(3x)
fx(x,y)=
fy(x,y)=
fxx(x,y)=
fxy(x,y)=
fyy(x,y)=
part 5)
For the function given below, find the numbers (x,y) such that
fx(x,y)=0 and fy(x,y)=0
f(x,y)=6x^2+23y^2+23xy+4x−2
Answer: x= and...

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

Let fx,y (x,y) = 3 e^-(x+y) for 0 < x <1/2y and y>0. a)
Find f x(x) and f y( y) . b) Write out the integral
necessary to find , Fx,y ( u v) . DO NOT EVALUATE THE INTEGRAL.

a)Prove that the function
u(x, y) = x -y÷x+y
is harmonic and obtain a conjugate function v(x, y) such that
f(z) = u + iv is analytic.
b)Convert the integral
from 0 to 5 of (25-t²)^3/2 dt
into a Beta Function and evaluate the resulting function.
c)Solve the first order PDE
sin(x) sin(y)
∂u
∂x + cos(x) cos(y)
∂u
∂y = 0
such that u(x, y) = cos(2x), on x + y =
π
2

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