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76

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.

77

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 $\frac{a}{b+z}$.

78

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 $I$.

79

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 $|z|>5$.

80

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 $g(w)=f(1/w)$?).

81

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  $\arctan (z)$.

82

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  $\Gamma$.

83

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 $|z|>2$.

84

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 $I$.

85

Term Test 2 / TT2A Problem 3

« on: November 24, 2018, 04:55:52 AM »

Find all singular points of
$$
f(z)=(z^2-1)\cot(\pi z^2)
$$
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 $g(w)=f(1/w)$?).

86

Term Test 2 / TT2A Problem 2

« on: November 24, 2018, 04:55:21 AM »

(a) Find the decomposition into power series at ${z=0}$ of $$f(z)=(1-z)^{-1/3}.$$ What is the radius of convergence?

(b) Plugging in $z^2$ instead of $z$, integrating and multi[lying by $z^{-1}$, obtain a decomposition at $z=0$ of 
$$F(z)=\frac{1}{z}\int_0^z (1-z^2)^{-1/3}\,dz$$ (which is the special case of the generalized hypergeometric function).

87

Term Test 2 / TT2A Problem 1

« on: November 24, 2018, 04:54:06 AM »

Using Cauchy's integral formula calculate
$$
\int_\Gamma \frac{z\,dz}{z^2-4z+5},
$$
where $\Gamma$ is a counter-clockwise oriented simple contour, not passing through eiter
of $2\pm i$ in the following cases

(a) The point $2+i$ is inside  $\Gamma$ and $2-i$ is outside  it;

(b) The point $2-i$ is inside  $\Gamma$ and $2+i$ is outside it;

(c) Both points $2\pm i$ are inside  $\Gamma$.

88

Term Test 2 / No double-dipping!

« on: November 24, 2018, 04:52:23 AM »

Do not post solutions to more than one parallel problems for different sittings, f.e. TT2-P1 and TT2B-P1.

On the other hand, you may post solution to TT2-P1  and  participate in the discussion for TT2A-P1 and TT2B-P1.

89

Term Test 2 / TT2 Problem 5

« on: November 24, 2018, 04:50:29 AM »

Consider $$f(z)= \frac{5}{(z-2)(z+3)}$$ and decompose it into Laurent's series converging

(a) As $|z|<2$;

(b)  As $2<|z|<3$;

(c) As $|z|>3$.

90

Term Test 2 / TT2 Problem 4

« on: November 24, 2018, 04:48:36 AM »

$\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}$
Calculate an improper integral
$$
I=\int_0^\infty \frac{dx}{\sqrt{x}(x^2+1)}.
$$

Hint:

(a)  Calculate
$$
J_{R,\varepsilon} = \int_{\Gamma_{R,\varepsilon}} f(z)\,dz, \qquad f(z)=\frac{1}{\sqrt{z}(z^2+1)}
$$
where we have chosen the branch of $\sqrt{z}$ such that it is 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 $\int_{\gamma_R}  \frac{dz}{\sqrt{z}(z^2+1)}\to 0$ as $R\to \infty$ and $\int_{\gamma_\varepsilon}  \frac{dz}{\sqrt{z}(z^2+1)}\to 0$ as $\varepsilon\to 0^+$ where $\gamma_R$ and $\gamma_\varepsilon$ are large and small semi-circles on the picture. This will give you a value of
$$
\int_{-\infty}^0 f(z)\,dz + \int_0^{\infty} f(z)\,dz.
$$

(c) Express both integrals using $I$.