Topics for Exam II



Infinite Dimensional Spaces

a. Inner Product [pic], Orthogonality

b. Generalized Fourier Coefficients: [pic][pic]

c. Gram Schmidt Orthogonalization, [pic]

d. Legendre Polynomials – use of Rodrigues formula, three term recursion formula, generating function, and normalization - [pic]

e. Gamma Function [pic],

[pic][pic] [pic], [pic]

1. Complex Numbers

a. Know how to use polar forms [pic], [pic] and [pic]

b. [pic]

c. Complex Modulus and complex conjugate

d. nth roots [pic] for [pic]

e. Roots of Unity

2. Complex Functions

a. Determine real and imaginary parts of functions:[pic]

b. Complex functions [pic]etc.

3. Differentiation

a. Compute Derivative [pic]

b. Differentiability and CR Equations [pic]

c. Harmonic Functions: CR => [pic]and harmonic conjugate.

d. Terms - Holomorphic, Analytic, Entire, ….

4. Integration [pic]

a. Complex Path Integrals – parametrized over line segment, arcs, etc. [pic]

b. Path Independence, When can one deform contours?

c. Cauchy’s Theorem [pic]if [pic] is differentiable

d. Cauchy Integral Formula [pic]

e. Computing Residues

i. Res[pic] - simple poles

ii. Res[pic] - poles of order k

f. Residue Theorem [pic]

g. [pic], [pic]

h. Going from integrals over R to complex integrals

5. Series Expansions

a. Power series, Laurent series, Taylor series

b. Circle of convergence

c. Using geometric series

i. [pic],

ii. [pic]

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