 Fourier series: smoothness and decay of coefficients.
 Fourier transform (following chapter 18 of Zorich, Mathematical Analysis, 2).
 Motivation
 Examples
 Properties
 Kotelnikov formula
 Asymptotics of integrals (chapter 19 of ibid.)
 Laplace method
 Localization principle
 Standard integrals
 Applications: Stirling formula
 Tauberian theorems: regularly varying functions
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Math 487, Homework 5
\(\def\Res{\mathtt{Res}}\)
 Find function with Laplace transform equal to \[\frac{s^3 – 1}{(1 + s)^4}.\]
 Find Fourier coefficients of the function \(t^3\pi^2 t\) on \([\pi,\pi]\).
Use Parseval’s identity to find \[\sum_{n=1}^\infty 1/n^6.\]  Find general formula for the coefficients of the generating function \(W(z)\) given by \(W=1+zW^4\).
 Find asymptotic frequency of the numbers \(n\) such that the first digits of \(2^n\) and \(3^n\) coincide.
Math 487, Nov. 30
 Inverting Fourier transform.
 “Uncertainty principle”.
 Introduction to duality – asymptotics vs. local behavior
Math 487, Nov. 28
 Fourier series
 Elements of Fourier transform
 Parseval theorem.
Math 487, Nov. 26
 Using residues to estimate coefficients of generating functions
 Algebraic generating functions: Catalan numbers
Exercises:
 Using residues, find coefficients of generating function \(H(z)=1+z+3z^2+\ldots\) satisfying
$$
H(z)=1+zH(z)^3.
$$  Consider linear recurrence given by
\[
f_n+f_{n1}+\ldots+f_{nk+2}+f_{nk+1}=0.
\]
Solve it (i.e. find the general formula for its terms \(f_n\)).
Math 487, Nov. 16
 Rational generating functions and linear recurrences
 Expansions into elementary fractions and coefficients of rational generating functions
Math 487, Nov. 14
 Rouche theorem.
 Generating functions, examples.
Math 487, homework 4
\(\def\Res{\mathtt{Res}}\)
 Which of the following functions is complex analytic in some region? (here \(z=x+iy\)):
 \(x+iy\mapsto (3x^2yy^3)+i(x^33xy^2)\);
 \(z\mapsto x^2+i y^2\).
 \(x+iy\mapsto i(3x^2yy^3)+(x^33xy^2)\);
 \(x+iy\mapsto (yix)/(x^2+y^2)\);
 Sketch the image of the circles \(z=1/3; z=3\) under the mapping \(z\mapsto z^3\bar{z}\).
 Find the integral of
$$
\frac{dz}{z^48z^29}
$$
over the circles of radii 2 and 4, each oriented counterclockwise.  Find
 \[\int_0^\infty \frac{z^2 dz}{(1+z^2)^3}.\]
 \[\int_0^\infty \frac{log(z)dz}{1+z^3}.\]
Solution:We start with finding \[I_0:=\int_0^\infty \frac{dz}{1+z^3}.\]
Consider the form \(\frac{\log(z)dz}{1+z^3}\) where we take the branch of \(\log\) having imaginary part 0 on the positive real axis and extending it counterclockwise to the plane so that it comes back to the positive real axis picking an extra \(2\pi i\).
Math 487, Nov. 12
 Fresnel integral.
 Logarithmic residues. Rouche theorem.
Math 487, midterm 2
$latex \def\Comp{\mathbb{C}}\def\Real{\mathbb{R}}$
Midterm 2.

 [15] Let $latex u_1, u_2, u_3\in\Real^n$ are three unit vectorcolumns such that the angle between any two of them is $latex 120^\circ$. Find the spectrum of the $latex n\times n$ matrix
$$
A=u_1\cdot u_1^*+u_2\cdot u_2^*+u_3\cdot u_3^*
$$
(here $latex u*$ is the transposition of $latex u$, i.e. a vectorrow).  [15] Find curl and div for the following vector fields in $latex \Real^3$:
 $latex B=(yz, xz, xy)$;
 $latex C=(y,z,x)$;
 $latex D=(x^3,y^3,z^3)$.
 [20] Find the integral of the 2form $latex \omega^2_D$, where the vector field $latex D$ is defined in the previous problem, over the surface of unit sphere in $latex \Real^3$.
 [15] Let $latex u_1, u_2, u_3\in\Real^n$ are three unit vectorcolumns such that the angle between any two of them is $latex 120^\circ$. Find the spectrum of the $latex n\times n$ matrix