Archive | math 487 fa18

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

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.

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
$f_n+f_{n-1}+\ldots+f_{n-k+2}+f_{n-k+1}=0.$
Solve it (i.e. find the general formula for its terms $$f_n$$).

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

$$\def\Res{\mathtt{Res}}$$

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 $$2\pi i$$.

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.

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 $latex \Real^3$.