Question

Verify the Caucy-riemann equations for the functions u(x,y), v(x,y) defined in the given domain

- u(x,y)=x³-3xy², v(x,y)=3x²y-y³, (x,y)ɛR
- u(x,y)=sinxcosy,v(x,y)=cosxsiny (x,y)ɛR
- u(x,y)=x/(x²+y²), v(x,y)=-y/(x²+y²),(x²+y²), ( x²+y²)≠0
- u(x,y)=1/2 log(x²+y²), v(x,y)=sin¯¹(y/√¯x²+y²), ( x˃0 )

In each case,state a complex functions whose real and imaginary parts are u(x,y) and v(x,y)

Answer #1

Two functions, u(x,y) and v(x,y), are said to verify the
Cauchy-Riemann
differentiation equations if they satisfy the following
equations ∂u\dx=∂v/dy and ∂u/dy=−(∂v/dx)
a. Verify that the Cauchy-Riemann differentiation equations can
be written in the polar coordinate form as
∂u/dr=1/dr ∂v/dθ and ∂v/dr =−1/r ∂u/∂θ
b. Show that the following functions satisfy the Cauchy-Riemann
differen- tiation equations
u=ln sqrt(x^(2)+y^(2)) and v= arctan y/x.

Can you find a functionv (x,y) so that u+iv is entire with
u(x,y) =x^3+ 3xy^2 ? (cauchy- riemann equations---hint)

Solve the following nonhomogenous Cauchy-Euler equations for x
> 0.
a. x^(2)y′′+3xy′−3y=3x^(2).

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

If z=(x+4y)ex+y,x=ln(u),y=v,z=(x+4y)ex+y,x=ln(u),y=v, find
∂z∂u∂z∂u and ∂z∂v∂z∂v. The variables are restricted to domains on
which the functions are defined.

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.

Suppose u(t,x) and v(t,x ) is C^2 functions defined on R^2 that
satisfy the first-order system of PDE Ut=Vx,
Vt=Ux,
A.) Show that both U and V are classical solutions to the wave
equations Utt= Uxx.
Which result from multivariable calculus do you need to justify
the conclusion.
B)Given a classical sol. u(t,x) to the wave equation, can you
construct a function v(t,x) such that u(t,x), v(t,x)
form of sol. to the first order system.

8) Suppose a consumer’s utility function is defined by
u(x,y)=3x+y for every x≥0 and y≥0 and
the consumer’s initial endowment of wealth is w=100. Graphically
depict the income and
substitution effects for this consumer if initially Px=1 =Py and
then the price of commodity x
decreases to Px=1/2.

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

if y=uv, where u and v are functions of x, show that the nth
derivative of y with respect to x is given by
(also known as Leibniz Rule)

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