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### Topics - Victor Ivrii

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31
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 5A
« on: November 27, 2018, 07:20:10 AM »
Determine the number of zeros of
$$2z^5 + 4z + 1.$$
(a) in the disk $\{z\colon |z|<1\}$;

(b) in the annulus $\{z\colon 1 <|z| < 2\}$.

(c) in the domain $\{z\colon |z|>2\}$.

Show that they are all distinct.

32
##### End of Semester Bonus--sample problem for FE / Readme
« on: November 27, 2018, 04:37:40 AM »
I will post here sample problems for FE. I will post just one version for problems of the type, covered by one of TT, and two versions for problems of the type, not covered by one of TT.

33
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 6
« on: November 27, 2018, 03:57:44 AM »
Calculate
$$\int_0^\infty \frac{x\sin (x)}{1+x^4}.$$

Hint:
Consider
$$\int _\Gamma f(z)\,dz \qquad \text{with } \ f(z)=\frac{ze^{iz}}{1+z^4}$$
over contour $\Gamma$ on the picture below:

34
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 5
« on: November 27, 2018, 03:57:31 AM »
Show that the equation
$$e^{z}=e^2z$$
has a real root in the unit disk $\{z\colon |z|<1\}$.

Are there non-real roots?

35
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 4
« on: November 27, 2018, 03:57:15 AM »
(a) Find the Mobius's transformation $f(z)$ mapping the unit disk $\{z\colon |z|<1\}$ onto exterior $\{w\colon |w|>1\}$ of the unit disk, such that $f(0)=5$ and $f(-1)=-1$.

(b) Find the fixed points of $f$ (points s.t. $f(z)=z$).

(c) Find the stretch ($|f'(z)|$) and the rotation angle ($\arg(f'(z))$) of $f$ at $z$.

36
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 3
« on: November 27, 2018, 03:56:33 AM »
Find all singular points, classify them, and find residues at these points of
$$f(z)= \frac{\cos(z/6)}{\sin^2(z)} + \frac{z}{\sin(z)}.$$
infinity included.

37
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 2
« on: November 27, 2018, 03:51:47 AM »
(a) Consider map
$$z\mapsto w=f(z):=\cos(z).$$
(b) Check that lines $\{z\colon \Im z =q \}$  are mapped onto confocal ellipses  $\{w=u+iv\colon \frac{u^2}{a^2}+\frac{v^2}{b^2}=1\}$ with $a^2-b^2=1$ and find $a=a(q)$ and $b=b(q)$.

(c) Check that lines $\{z\colon \Re z =p \}$ are mapped onto confocal  hyperbolas  $\{w=u+iv\colon \frac{u^2}{A^2}+\frac{v^2}{B^2}=1\}$ with $A^2+B^2=1$ and find $A=A(p)$ and $B=B(p)$.

(d) Find to what domain this  function  maps the strip  $\mathbb{D}=\{z\colon 0<\Re p < \pi\}$.

(e) Draw both domains.

(f) Check if the correspondence is one-to-one.

38
##### End of Semester Bonus--sample problem for FE / FE Sample--Problem 1
« on: November 27, 2018, 03:51:11 AM »
(a) Decompose into Taylor series at $0$ function
$$f(z)=\frac{1}{z^2-8z+25}.$$

Find the radius of convergence $r$. Determine if the series is converging at $|z|=r$ (consider all points $z$ satisfying $|z|=r$).

(b) Decompose into Laurent's series at $\infty$ the same function. Also find the radius $R$ (so it converges as $|z|> R$).
Determine if the series is converging at $|z|=R$ (consider all points $R$ satisfying $|z|=R$).

Hint:
Represent $f(z)$ as the sum of functions of the form $\frac{a}{b+z}$.

39
##### Term Test 2 / TT2B Problem 4
« on: November 24, 2018, 05:25:15 AM »
Calculate an improper integral
$$I=\int_0^\infty \frac{\ln(x)\sqrt{x}\,dx}{(x^2+1)}.$$

Hint:

(a) Calculate
$$J_{R,\varepsilon} = \int_{\Gamma_{R,\varepsilon}} f(z)\,dz, \qquad f(z)=\frac{\sqrt{z}\log(z)}{(z^2+1)}$$
where we have chosen the branches of $\log(z)$ and $\sqrt{z}$ such that they are analytic on the upper half-plane $\{z\colon \Im z>0\}$ and is real-valued for $z=x>0$. $\Gamma_{R,\varepsilon}$ is the contour on the figure below:

(b)  Prove that $\int_{\gamma_R} \frac{\sqrt{z}\log(z)\,dz}{(z^2+1)}\to 0$ as $R\to \infty$ and $\int_{\gamma_\varepsilon} \frac{\sqrt{z}\log(z)\,dz}{(z^2+1)}\to 0$ as $\varepsilon\to 0^+0$ where $\gamma_R$ and $\gamma_\varepsilon$ are large and small semi-circles on the picture. This will give you a value of
$$\int_{-\infty}^0 f(z)\,dz + \int_0^{\infty} f(z)\,dz. \tag{*}$$

(c) Express both integrals using $I$.

40
##### Term Test 2 / TT2B Problem 5
« on: November 24, 2018, 05:22:35 AM »
Consider $$f(z)= \frac{8}{(z-3)(z+5)}$$ and decompose it into Laurent's series converging

(a) As $|z|<3$;

(b) As $3<|z|<5$;

(c) As $|z|>5$.

41
##### Term Test 2 / TT2B Problem 3
« on: November 24, 2018, 05:21:14 AM »
Find all singular points of
$$f(z)=\frac{\sin (\pi z)}{\sin(\pi z^3)}$$
and determine their types (removable, pole (in which case what is it's order), essential singularity, not isolated singularity, branching point).

In particular, determine singularity at $\infty$ (what kind of singularity we get at $w=0$ for $g(w)=f(1/w)$?).

42
##### Term Test 2 / TT2B Problem 2
« on: November 24, 2018, 05:20:09 AM »
(a) Find the decomposition into power series at ${z=0}$ of $$f(z)=(1-z)^{-1}.$$ What is the radius of convergence?

(b) Plugging in $-z^2$ instead of $z$ and integrating, obtain a decomposition at $z=0$ of  $\arctan (z)$.

43
##### Term Test 2 / TT2B Problem 1
« on: November 24, 2018, 05:19:15 AM »
Using Cauchy's integral formula calculate
$$\int_\Gamma \frac{dz}{z^2-6z+25},$$
where $\Gamma$ is a counter-clockwise oriented simple contour, not passing through eiter
of $1\pm 3i$ in the following cases

(a) The point $3+4i$ is inside  $\Gamma$ and $3-4i$ is outside  it;

(b) The point $3-4i$ is inside  $\Gamma$ and $3+4i$ is outside it;

(c) Both points $3\pm 4i$ are inside  $\Gamma$.

44
##### Term Test 2 / TT2A Problem 5
« on: November 24, 2018, 05:17:20 AM »
Consider $$f(z)= \frac{3z}{(z-2)(z+1)}$$ and decompose it into Laurent's series converging

(a) As $|z|<1$;

(b) As $1<|z|<2$;

(c) As $|z|>2$.

45
##### Term Test 2 / TT2A Problem 4
« on: November 24, 2018, 05:15:23 AM »
Calculate an improper integral
$$I=\int_0^\infty \frac{\sqrt{x}\,dx}{(x^2+2x+2)}.$$

Hint:

(a) Calculate
$$J_{R,\varepsilon} = \int_{\Gamma_{R,\varepsilon}} f(z)\,dz, \qquad f(z)=\frac{\sqrt{z}}{(z^2+2z+2)}$$
where we have chosen the branch of $\sqrt{z}$ such that it is analytic inside  $\Gamma$ and is real-valued for $z=x+i0$ with $x>0$. $\Gamma=\Gamma_{R,\varepsilon}$ is the contour on the figure below:

(b) Prove that $\int_{\gamma_R} \frac{\sqrt{z}\,dz}{(z^2+1)}\to 0$ as $R\to \infty$ and $\int_{\gamma_\varepsilon} \frac{\sqrt{z}\,dz}{(z^2+1)}\to 0$ as $\varepsilon\to 0^+$ where $\gamma_R$ and $\gamma_\varepsilon$ are large and small circles on the picture. This will give you a value of
$$\int_{\infty}^0 f(x-i0)\,dx + \int_0^{\infty} f(x+i0)\,dx \tag{*}$$
where $f(x\pm i0)=\lim _{\delta\to 0^+} f(x+i\delta)$.

(c) Express both integrals using $I$.

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