Question

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

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) =?

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

Problem 7. Consider the line integral Z C y sin x dx − cos x
dy.
a. Evaluate the line integral, assuming C is the line segment
from (0, 1) to (π, −1).
b. Show that the vector field F = <y sin x, − cos x> is
conservative, and find a potential function V (x, y).
c. Evaluate the line integral where C is any path from (π, −1)
to (0, 1).

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

Solve the following initial/boundary value problem:
∂u(t,x)/∂t = ∂^2u(t,x)/∂x^2 for t>0, 0<x<π,
u(t,0)=u(t,π)=0 for t>0,
u(0,x)=sin^2x for 0≤x≤ π.
if you like, you can use/cite the solution of Fourier sine
series of sin^2(x) on [0,pi] = 1/4-(1/4)cos(2x)
please show all steps and work clearly so I can follow your
logic and learn to solve similar ones myself.

1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

Evaluate the scalar surface integral ∬ S1 (x) dS, where S1 is
the portion of the helicoid which is the image of the
parametrization r( u , v ) =< u cos( v ) , u sin( v ) , v >
over 0 ≤ u ≤ 2 and 0 ≤ v ≤ π/2.

Find the general solution of uxx − 3uxy +
2uyy = 0 using the the method of characteristics: let v
= y + 2x and w = y + x; define U(v, w) to be U(v, w) = U(y + 2x, y
+ x) = u(x, y); derive and solve a PDE for U(v, w); convert back to
u(x, y).

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?

Set up the triple integral, including limits, of the function
over the region.
f(x, y, z) = sin z, x ≥ 0, y ≥ 0, and below the plane 2x + 2y +
z = 2

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