Prove that │∑ zn │ ≤ ∑│zn │ where zn is a complex number.

Functions of a Complex Variables ∞ ∞ Prove that │∑ zn │ ≤ ∑│zn │ where zn is a complex number. n =1 n =1 ...continues

Prove that (a) │ z1 │-│ z2 │ ≤ │z1 - z2│ ≤ │ z1 │+ │ z2 │ (b) │ z1 │-│ z2 │ ≤ │z1 + z2│ ≤ │ z1 │+│ z2 │ where z1 , z2 are complex numbers.

Functions of a Complex Variables Prove that: (a) │ z1 │-│ z2 │ ≤ │z1 - z2│ ≤ │ z1 │+ │ z2 │ (b) │ z1 │-│ z2 │ ≤ │z1 + z2│ ≤ │ z1 │+│ z2 │ ...continues

Partial Induction Proof of Cauchy's Integral Formula

see attached file...it is a full induction proof of Cauchy Integral Formula, with the base case step missing. All I have to do is show that it holds for "n=1", using the rest of the proof as an example...however i am having trouble showing it.

If u = sin x . cosh y + 2cos x . sinh y + x2 – y2 + 4xy , then prove that u is a harmonic function and find the analytic function f(z) = u + iv .

Functions of a Complex Variables Analytic Functions If u = sin x . cosh y + 2cos x . sinh y + x2 – y2 + 4xy , then prove that u is a harmonic function and find the analytic funct ...continues

Complex Variables

If a > e prove that the equation a*z^n=e^z has n solutions (counting multiplicities) inside of the circle |z|=1.

Analyticity Proof

Suppose that f: C->C and that f is analytic at a point z0 element of C. Prove that there exists a real number r>0 such that, the nth derivative of z0=[n!/(2 pi r^n)]x[int(e^(-niy)f(z0+re^(iy)) from 0 to 2pi with respect to y for all n element of Natural numbers.

Laurent Series

Find the Laurent series about all singular points of f(z) = 1/(z(z+1)^2) {see attachment} Thanks.

Mapping (Quarter-Plane; Half-Line)

Find the image of the quarter-plane {see attachment} under the mapping {see attachment}. Show graphs (shaded regions) in the w-plane and identify the images of the half-lines {see attachment}.

Computing an Integral Using Residues

Use residue theory to compute the integral: (see attachment)

Classifying Isolated Singularities

Classify all the isolated singularities of the following functions (classify as removable, pole of order m, or essential). Explain the reasoning for each classification. *See attachment for functions*