Question

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?

Answer #1

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.

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

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.

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

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

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) =?

Numerical analysis problem, please show all steps thank you.
A four times continuously differentiable function f is given by
the following data: f(1.1)=2, f(1.3)=1.5, f(1.5)=1.2, f(1.7)=1.6.
Assume that |f ''''(t)|=<100 for 1.1<t<1.7. Find the
estimate for f '' (1.3). Give an error bound.

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

Please show all steps, thank you.
Using the Taylor formula for f(x+h) and f(x-h) with f ''' in the
error term, find the error of the approximate formula f '' (x) =
(f(x+h)+f(x-h)-2f(x))/(h^2) in terms of f ''' (eta), for some point
eta between x-h and x+h. Then give an upper bound for the absolute
error assuming that |f ''' (t)| =< M for t between x-h and
x+h.

Please show steps for solving all parts. im really confused,
thank you!
Calculate E, v, & r for Z=1 (Hydrogen), n=1 & 2; and E,
v, & r for Z=2(Helium), n=1

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