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Math 487, week of Dec 3

  • 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

\(\def\Res{\mathtt{Res}}\)

  1. Find function with Laplace transform equal to \[\frac{s^3 – 1}{(1 + s)^4}.\]
  2. Find Fourier coefficients of the function \(t^3-\pi^2 t\) on \([-\pi,\pi]\).
    Use Parseval’s identity to find \[\sum_{n=1}^\infty 1/n^6.\]
  3. Find general formula for the coefficients of the generating function \(W(z)\) given by \(W=1+zW^4\).
  4. Find asymptotic frequency of the numbers \(n\) such that the first digits of \(2^n\) and \(3^n\) coincide.
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Math 487, Nov. 30

  • Inverting Fourier transform.
  • “Uncertainty principle”.
  • Introduction to duality – asymptotics vs. local behavior
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Math 487, Nov. 28

  • Fourier series
  • Elements of Fourier transform
  • Parseval theorem.
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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
    \[
    f_n+f_{n-1}+\ldots+f_{n-k+2}+f_{n-k+1}=0.
    \]
    Solve it (i.e. find the general formula for its terms \(f_n\)).
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Math 487, Nov. 16

  • Rational generating functions and linear recurrences
  • Expansions into elementary fractions and coefficients of rational generating functions
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Math 487, Nov. 14

  • Rouche theorem.
  • Generating functions, examples.
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Math 487, homework 4

\(\def\Res{\mathtt{Res}}\)

  1. Which of the following functions is complex analytic in some region? (here \(z=x+iy\)):
    1. \(x+iy\mapsto (3x^2y-y^3)+i(x^3-3xy^2)\);
    2. \(z\mapsto x^2+i y^2\).
    3. \(x+iy\mapsto i(3x^2y-y^3)+(x^3-3xy^2)\);
    4. \(x+iy\mapsto (y-ix)/(x^2+y^2)\);
  2. Sketch the image of the circles \(|z|=1/3; |z|=3\) under the mapping \(z\mapsto z^3-\bar{z}\).
  3. Find the integral of
    $$
    \frac{dz}{z^4-8z^2-9}
    $$
    over the circles of radii 2 and 4, each oriented counterclockwise.
  4. Find
    1. \[\int_0^\infty \frac{z^2 dz}{(1+z^2)^3}.\]
    2. \[\int_0^\infty \frac{log(z)dz}{1+z^3}.\]
      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 \(2\pi i\).

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Math 487, Nov. 12

  • Fresnel integral.
  • Logarithmic residues. Rouche theorem.
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Math 487, midterm 2

$latex \def\Comp{\mathbb{C}}\def\Real{\mathbb{R}}$

Midterm 2.

    1. [15] Let $latex u_1, u_2, u_3\in\Real^n$ are three unit vector-columns 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 vector-row).
    2. [15] Find curl and div for the following vector fields in $latex \Real^3$:
      1. $latex B=(yz, xz, xy)$;
      2. $latex C=(y,z,x)$;
      3. $latex D=(x^3,y^3,z^3)$.
    3. [20] Find the integral of the 2-form $latex \omega^2_D$, where the vector field $latex D$ is defined in the previous problem, over the surface of unit sphere in $latex \Real^3$.
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