Laurent series for a complex-valued function

Consider f(z) = [(z-i)(z+4)(z-3)]^(-1) restricted to the domain of definition 0 < |z|< infinity How many different Laurent series centered at z_0 = 0 does it have? Explain. Discuss the convergence and divergence sets of each of those Laurent series. Find two non-zero terms of the Laurent series which represents this ...continues

Poles & Singularities

Classify the behavior at infinity (analytic, pole, zero, or essential singularity; if a zero or pole, give its order) of the following functions: f1(z) = (z^3 + i)/z f2(z) = e^(tan(1/z))

Use residues to evaluate the integral of a trigonometric function.

Use residues to evaluate integral of a trigonometric function. Please see the attached file for the fully formatted problems.

Complex Analysis : Evaluating integrals using residue theorem

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Taylor Series Representation and Residue

Write f(z):=16z/(z^2 +1)^3 as f(z)= h(z)/(z - i)^3 with an explicit expression for the function h(z). Explain why h(z) has a Taylor series representation about i and use this representation to find explicitly the principal part of f at i. Hence, find the numerical value of the residue of f at i. Please see t ...continues

Complex Analysis : Evaluating real integrals using the residue theorem.

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Evaluating integrals using residue theorem

Please see the attached file for the fully formatted problems.

What is the fifth term in the following sequence?

What is the fifth term in the following sequence? asubcript n =n+asubscript n-1. if a1 equals -2, for n greater than or equal to 2.

Jordan's Lemma and Loop Integrals

Without evaluating the improper integrals and find the numerical value q of their quotient by considering the loop integral where is the semi-circular loop indented at the origin. Explain why Jordan's Lemma (see below) is inadequate here, and write a complete formulation of a more general Jordan's lemma ...continues

Harmonic Function: Analyticity, Compactness and Minimum Value

Let (attached) be a function that is analytic and not constant throughtout a bounded domain (attached) and continuous (attached) on its boundary (here domain is an open connected set). Prove, by considering (attached) , that the component function (attached) has a minimum value in the compact region (attached) which occurs on ...continues