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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