Question

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

Answer #1

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) =?

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?

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

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.

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.

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

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.

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

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...

Find the domain of the following function. Explain your
reasoning and express your answer in interval notation.
f(x)= ln (x)/ (e^2x) -3

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