Question

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) =?

Answer #1

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

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

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

Evaluate the following.
f(x, y) = x + y
S: r(u, v) = 5
cos(u) i + 5 sin(u)
j + v k, 0 ≤ u
≤ π/2, 0 ≤ v ≤ 3

4.Given F(x,y,z)=(cos(y))i+(sin(y))j+k, find divF and curlF at
P0(π/4,π,0) divF(P0)=? curlF(P0)= ?

4. Given L {sinh (-4t)} = -(4/ (s2 -16)), find L{ t
sinh (-4t) }
5. Given y''(t) + 3y(t) = e-2t cos t and y(0) = -1,
y'(0), find L { y(t) }

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?

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.

Let S be the portion of the surface z=cos(y) with 0≤x≤4 and
-π≤y≤π. Find the flux of F=<e^-y,2z,xy> through S:
∫∫F*n dS

Consider the function f(x, y) = 3e^(2y)cos x.
(a) Find the value of the direction derivative of f at the point
(1, 0) in the direction
of the point (2, 1).
(b) Find the direction of maximum increase of f from the point (π,
1).
Find the rate of that max increase.

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