1.2 Topics covered

This is list of lectures and topics covered in each

Table 1.1: Topics covered
#	Date	Topics
1	Wed. Sept 4, 2018	1. Infinite series, geometric series. 2. conditions for convergence, 3. harmonic series, alternating series. 4. tests for convergence such as ratio test, integral test.
2	Monday Sept 10, 2018	1. Talked about series solution to \((1-x^2)y''-2 x y'+ n(n+1) y=0 \). 2. Binomial series \((1+x)^r = 1+ r x + \frac{r(r-1) x^2}{2!} + \dots \) 3. series of \(e,\sin (x),\cos (x)\). 4. Introducing Bernoulli numbers. 5. Show that alternating series is convergent but not absolutely. 6. Leibniz condition for convergence and its proof. 7. Showed that sum of \(1-1/2+1/3-1/4+\dots \) is \(\ln 2\). 8. Working with absolutely convergent series.
3	Wed Sept 12, 2018	1. Familiar series, \(e,\sin x,\cos x,\ln (1+x),\arctan (x)\) 2. How to get Bernulli numbers. More on Bernulli numbers but I really did not understand these well and how to use them. hopefully they will not be on the exam. 3. Started Complex analysis. Basic introduction. Properties of complex numbers and mapping.
4	Monday Sept 17, 2018	1. complex functions \(u(x,y)+i v(x,y)\) 2. continuity in complex domain. 3. Derivative in complex domain and how direction is important. 4. Cauchy-Riemman equation to test for analytical function. 5. Harmonic functions. Exponential function in complex domain. 6. Multivalued functions, such as \(\log z\). 7. How to obtain inverse trig function and solve \(w=\arcsin (z)\)
5	Wed Sept 19, 2018	1. derivative in complex plane. Definition of analytic function. 2. \(\log (z)\) and \(\sqrt (z)\) in complex plane and multivalued. Branch points and branch cuts. 3. Integration over contour. Parameterization \(\int _C f(z)\, dz=\int _a^{b} f(z(t)) z'(t)\, dt\) 4. Cauchy-Goursat theorem: \(\oint f(z)=0\) for analytical functions. Proof using Cauchy-Riemman equations and Green theorem. 5. Cauchy integral formula \(2 \pi i f(z_0) = \oint \frac{f(z)}{z-z_0} \,dz\) 6. Like in real, in complex domain, Continuity Does Not Imply Differentiability. 7. More on analytic functions and multivalued functions. Principal value. 8. Power functions \(z^p=e^{p \ln z}\) 9. Complex integration.
6	Monday Sept 24, 2018	1. Proof of Cauchy integral formula. 2. Maximum moduli of analytic functions. If \(f(z)\) is analytic in \(D\) and not constant, then it has no maximum value inside \(D\). The maximum of \(f(z)\) is on the boundary. 3. Taylor series for complex functions and Laurent series.
7	Wed Sept 26, 2018	1. If number of terms in principal part of Laurent series is infinite, then it is essential singularity. 2. Proof of Laurent theorem. 3. properties of power series. Uniqueness. 4. Residues, types of singularities. How to find residues and examples
8	Monday Oct 1, 2018	1. Residue theorem \(\oint _C f(z)\, dz= 2 \pi i \sum \text{residues inside C}\) 2. examples using Residue theorem. 3. Analytic continuation. Examples. 4. \(\Gamma (z)\) function. Defined for \(\Re (z)>0\). Using analytical continuation to extend it to negative complex plane. Euler representation and Weistrass represenation.
9	Wed Oct 3, 2018	1. More on Euler represenation of \(\Gamma (z)\) and how to use it for extending definition \(\Gamma (z)=\int _0^{\infty } e^{-t} t^{z-1}\,dt\) for negative \(z\) using \(\Gamma (z)=\frac{\Gamma (z+1)}{z}\) for \(-1<z\). 2. Euler reflection formula \[ \Gamma (x)\Gamma (1-x)=\int _0^{\infty } \frac{t^{x-1}}{1+t}\,dt = \frac{\pi }{\sin (\pi x)} \] 3. proof of Euler reflection formula using contour integration. 4. Some useful formulas for \(\Gamma (z)\) 5. Method for integrations, some tricks to obtain definite integrations.
10	Monday Oct 8, 2018	No class.
11	Wed Oct 10, 2018	No class.
12	Monday Oct 15, 2018	1. More on method of integration. Starting Contour integration. 2. How to decide that \(\int _{C_R} f(z)=0\) on the upper half plane. Using Jordan inquality. 3. More examples of integrals on real line using contour integration.
13	Wed Oct 17, 2018	1. More contour integrations. 2. Starting approximation expansion of integrals. Example using error function \(\erf (x)=\frac{2}{\sqrt \pi } \int _0^{x} e^{-t^2}\,dt\) by applying Taylor series. 3. Large \(x\) expansion by repeated integration by parts. 4. Starting Asymprtotic series. Definition. Example on finding \(S(x)\) for \(\erf (x)\) for large \(x\). When to truncate. 5. Saddle point methods of integration to approximate integral for large \(x\).
14	Monday Oct 22, 2018	1. More saddle point integration. 2. Saddle point methods of integration to approximate integral for large \(x\). Method of steepest decsent. Example to find \(\Gamma (x+1)=\int _0^{\infty } t^x e^{-t}\,dt = \sqrt{2 \pi x} x^x e^{-x}\) 3. extend saddle point method to complex plane. Finding correct angle. Long example.
15	Wed Oct 24, 2018	1. More on saddle point in complex plane. Angles. Example applied on \(\int \Gamma (1+z)=\int _0^\infty \exp{-t+ z \ln t}\, dt\) 2. how to determine coefficients of asymptotic series expansion. 3. Starting new topic. Fourier series. Definitions. 4. proprties of Fourier series. Examples how to find \(A_n,B_n\).
16	Friday Oct 26, 2018	Make up lecture. 1. More on Fourier series. Examples. Fourier series using the complex formula. 2. Parseval identity. 3. Fourier Transform derivation.
17	Monday Oct 29, 2018	1. Fourier transform pairs. 2. How to find inverse fourier transform. Generalization to higher dimensions. 3. Properties of Fourier transform. 4. convolution. 5. Example on driven harmonic oscillator. 6. Statring ODE’s. Order and degree of ODE.
18	Wed Oct 31, 2018	1. More on first order ODE’s. Separable, exact. How to find integrating factor. 2. Bernulli ODE \(y'+f(x) y = g(x) y^n\) 3. Homogeneous functions. defintion. order of. 4. isobaric ODE’s.
19	Monday Nov 5, 2018	1. How to find integating factor for exact ODE. 2. Finished example on isobaric first order ODE. \[ x y^2(3 y \,dx + x\, dy) - ( 2 y \, dx-x \, dy)=0 \] 3. Higher order ODE’s. How to solve. How to find particular solution. Undetermined coefficients. What to do if forcing function has same form as one of the solutions to homogeneous solutions. 4. How to use power series to solve nonlinear ode \(y''=x-y^2\)
20	Wed Nov 7, 2018	First exam
21	Monday Nov 12, 2018	1. More on higher order ODE’s. Series solutions. 2. ordinary point. Regular singular point. Example Legendre ODE \((1-x^2)y''-2 x y'+n(n+1) y=0\). 3. Example for regular singular point, Bessel ODE \(x^2 y''+ x y'+(x^2-m^2)y=0\) Use \(y=x^2 \sum _{n=0}^{\infty } c_n x^n\)
22	Wed Nov 14, 2018	1. Continue Bessel ODE \(x^2 y''+ x y'+(x^2-m^2)y=0\) solving using \(y=x^2 \sum _{n=0}^{\infty } c_n x^n\). How to find second independent solution.
23	Monday Nov 19, 2018	1. Started on Sturm Lioville, Hermetian operators 2. setting Bessel ODE in Sturm Lioville form 3. more on Hermitian operator. 4. Wronskian to check for linear independece of solutions.
24	Wed Nov 21, 2018	Thanks Giving.
25	Monday Nov 26, 2018	1. finding second solution to Bessel ODE for \(m\) integer using the Wronskian. \(W(x)=\frac{C}{p(x)}=\frac{-2 \sin{\pi m}}{\pi }\) 2. Generating functions to find way to generate Besself functions.
26	Wed Nov 28, 2018	1. Using Generating functions 2. Bessel functions of half integer order, spherical Bessel functions 3. Legendre polynomials, recusrive relations. 4. orthonomalization. 5. physical applications
27	Monday December 3, 2018	1. Second solution to Legendre using Wronskian 2. Spherical harmonics 3. Normalization of eigenfunctions 4. Degenerncy, using Gram-Schmidt to find other L.I. solutions. 5. Expanding function using complete set of basis functions, example using Fourier series 6. Inhomogeneous problems, starting Green function
28	Wed December 5, 2018	1. Green function. Solution to the ODE with point source. 2. Example using vibrating string \(y''+k^2 y=0\). Find Green function. Two Methods. Use second method. 3. Started on PDE.
29	Friday December 7, 2018	(Make up lecture) 1. more PDE’s. Solve wave PDE in 1D 2. separation of variables. Solve Wave PDE in 3D in spherical coordiates.
30	monday December 10, 2018	1. Solving wave PDE in 3D in spherical coordinates. Normal modes. 2. Solving wave PDE in 3D in cylindrical coordinates. 3. Inhomogeneous B.C. on heat PDE. Break it into 2 parts.
31	Wed December 12, 2018	Last lecture. 1. Finish Inhomogeneous B.C. on heat PDE. Break it into 2 parts. Final solution, using Fourier series. 2. Last problem. INtegral transform method. Solving heat pde on infinite line using Fourier transform.