Question

discuss the analyticity of the function using the Cauchy-Riemann equations

w = log(z)

Answer #1

Prove that a function f(z) which is complex differentiable at a
point z0 satisfies the Cauchy-Riemann equations at that point.

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.

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

Apply Cauchy-Riemann to: f(z)=ln|z| +i Arg(z) What can you
conclude?
I'm not really even sure to start with this one and I'm just
really confused. Any help would be appreciated, thank you!

if x= log (w) and y=log(z), what is log (wz^3) in terms of x and
y.

[Cauchy-Euler equations] For the following equations with the
unknown function y = y(x), find the general solution by changing
the independent variable x to et and re-writing the equation with
the new unknown function v(t) = y(et).
x2y′′ +xy′ +y=0
x2y′′ +xy′ +4y=0
x2y′′ +xy′ −4y=0
x2y′′ −4xy′ −6y=0
x2y′′ +5xy′ +4y=0.

Instructions: Approximate the following definite integrals using
the indicated Riemann sums.
1. Z 9 1 x 1 + x dx using a left-hand Riemann sum L4 with n = 4
subintervals.
2. Z 3 0 x 2 dx using a midpont Riemann sum M3 using n = 3
subintervals.
3. Z 3 1 f(x) dx using a right-hand Riemann Sum R4, with n = 4
subintervals

suppose we have the following productions functions:
Q=LaK1-a .............1
Z= a log L+ ( 1-a) log K...........2
where Q is output, L is labour , K is capital , Z is a log
transformation of Q and a is a positive constant . prove
that equation 1 is homogenous while equation 2 is not; but both
equations are homothetic, implying that a non-homogeneous function
can still be homothetic,

Is it possible for an entire function w=f(z) to map the z-plane
into the circle |w|<1? Fully justify your explanation.

Problem 1. The Cauchy distribution with scale 1 has following
density function
f(x) = 1 / π [1 + (x − η)^2 ] , −∞ < x < ∞.
Here η is the location and rate parameter. The goal is to find the
maximum likelihood estimator of η.
(a) Find the log-likelihood function of f(x)
l(η; x1, x2, ..., xn) = log L(η; x1, x2, ..., xn) =
(b) Find the first derivative of the log-likelihood function
l'(η; x1, x2,...

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