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Topics - Victor Ivrii

Pages: 1 2 [3] 4 5 ... 58
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 »
$\renewcommand{\Re}{\operatorname{Re}}
  \renewcommand{\Im}{\operatorname{Im}}$
(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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