Description
This course introduces students to the fundamental concepts and methods of applied mathematics. The course consists of two parts: complex analysis and ordinary differential equations (initial value problems). The list of topics include:
• Complex Analysis: complex numbers, complex plane, Euler’s formula, regions in the complex plane, functions of complex variable, limits and continuity, stereographic projection, complex derivative, the Cauchy-Riemann equations, analytic functions, complex exponential, trigonometric functions, logarithmic function, branches, branch cuts, brunch points, power function, roots of unity, contour integrals, equivalence theorem, the Cauchy-Goursat theorem, deformation of paths, Cauchy’s integral formula, derivatives of analytic functions, Morera’s theorem, sequences and series, Taylor series, power series, circle of convergence, ratio test, Cauchy-Hadamard formula, continuity, integration, and analyticity of power series, Laurent series, zeros and singularities of analytic functions, residues, Cauchy’s residue theorem, improper integrals, Jordan’s lemma, analytic continuation.
• Ordinary Differential Equations (initial value problems): differential equations, general terminology, 1st order liners ODEs, 2nd order linear ODEs, existence and uniqueness, superposition principle, Wronskian, fundamental set of solutions, Abel’s theorem, reduction of order, variation of parameters, Green’s functions, the Laplace transform, applications to initial value problems, shifting theorems, initial value problems with discontinuous and impulsive forcing, convolution integral, the Mellin inversion formula, linear vs nonlinear ODEs, numerical methods for 1st order nonlinear IVPs, Euler’s method, backward Euler’s method, approximation errors, Heun’s method, Runge-Kutta methods, Adams methods, nst order nonlinear IVPs, power series solutions, Airy’s equations, ordinary and singular points, Fuchs’ theorem, Euler’s equations, the method of Frobenius.
• Complex Analysis: complex numbers, complex plane, Euler’s formula, regions in the complex plane, functions of complex variable, limits and continuity, stereographic projection, complex derivative, the Cauchy-Riemann equations, analytic functions, complex exponential, trigonometric functions, logarithmic function, branches, branch cuts, brunch points, power function, roots of unity, contour integrals, equivalence theorem, the Cauchy-Goursat theorem, deformation of paths, Cauchy’s integral formula, derivatives of analytic functions, Morera’s theorem, sequences and series, Taylor series, power series, circle of convergence, ratio test, Cauchy-Hadamard formula, continuity, integration, and analyticity of power series, Laurent series, zeros and singularities of analytic functions, residues, Cauchy’s residue theorem, improper integrals, Jordan’s lemma, analytic continuation.
• Ordinary Differential Equations (initial value problems): differential equations, general terminology, 1st order liners ODEs, 2nd order linear ODEs, existence and uniqueness, superposition principle, Wronskian, fundamental set of solutions, Abel’s theorem, reduction of order, variation of parameters, Green’s functions, the Laplace transform, applications to initial value problems, shifting theorems, initial value problems with discontinuous and impulsive forcing, convolution integral, the Mellin inversion formula, linear vs nonlinear ODEs, numerical methods for 1st order nonlinear IVPs, Euler’s method, backward Euler’s method, approximation errors, Heun’s method, Runge-Kutta methods, Adams methods, nst order nonlinear IVPs, power series solutions, Airy’s equations, ordinary and singular points, Fuchs’ theorem, Euler’s equations, the method of Frobenius.
General Information
| Name | Office Hours | |
|---|---|---|
Kostia Zuev | When? Where? | |
Allen Yang | When? Where? | |
Umran Serra Koca | When? Where? | |
Changhe Yang | When? Where? | |
Antonio Victor Machado de Oliveira | When? Where? | |
Keyu Wan | When? Where? | |
Anirudh Gajula | When? Where? | |
Kevin Gauld | When? Where? | |
Harshkooshal Gandhi | When? Where? | |
Jasmine Wang | When? Where? | |
Peicong Song | When? Where? | |
Kieran Vlahakis | When? Where? | |
Sujay Champati | When? Where? |