Math 198, Methods of Applied Mathematics I

Syllabus

1. Contour Integration

• Definite integrals of elementary functions
• Integral Representation of Functions
• Special Functions (Gamma,Bessel, Legendre, Hypergeometric)

2. Asymptotic Expansions

• Watson's Lemma
• Stationary Phase and Steepest Descent Methods
• Stoke's Phenomenon

3. Elementary Partial Differential Equations of Mathematical Physics

• Linear Evolution Equations
• Fourier Transform Solutions
• Phase Velocity and Group Velocity, Dispersion
• Long-time Asymptotics of Fourier Integral Solutions of PDE's

4. Special Functions from Physical Applications

• Elliptic functions
• Theta Functions

Elementary Bifurcation Theory

• Equilibrium Solutions and Linear Stability
• Local Bifurcations of Fixed Points of Vector Fields : Pitchfork, Transcritical, Saddle-Node, Hopf
• Structural Stability

Textbooks

Advanced Mathematical Methods for Scientists and Engineers, C. Bender and S. Orszag
Functions of a Complex Variable, G.F. Carrier, M. Krook and C.E. Pearson
Dynamics and Bifurcations, J.K. Hale and H. Kocak
A Course of Modern Analysis, E.T. Whittaker and G.N. Watson
Linear and Nonlinear Waves, G.B. Whitham