Question

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

Answer #1

discuss the analyticity of the function using the
Cauchy-Riemann equations
w = log(z)

Complex Analysis Proof - Prove: if f = u + iv is analytic in a
domain D, then u and v satisfy the Cauchy-Riemann equations in
D.

if
the function f is differentiable at a, prove the function f is also
continuous at a.

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!

Using the derivative definition, point the derivative value for
the given function f(z)=3/z^2 Find in Z0=1+i and write x+iy
algebraically.

PROVE USING IVT. Suppose f is a differentiable function on [s,t]
and suppose f'(s) > 0 > f'(t). Then there's a point p in
(s,t) where f'(p)=0.

Let f : R → R be a bounded differentiable function. Prove that
for all ε > 0 there exists c ∈ R such that |f′(c)| < ε.

Problem: Let y=f(x)be a differentiable function
and let P(x0,y0)be a point that is not on the graph of function.
Find a point Q on the graph of the function which is at a
minimum distance from P.
Complete the following steps. Let Q(x,y)be a point on the graph
of the function
Let D be the square of the distance PQ¯. Find an expression for
D, in terms of x.
Differentiate D with respect to x and show that
f′(x)=−x−x0f(x)−y0
The...

Consider the vector field F(x,y,z)=〈−3y,−3x,−4z〉. Find a
potential function f(x,y,z) for F which satisfies f(0,0,0)=0.

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