For questions about integration methods that use results from complex analysis and their applications

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2
votes
2answers
38 views

Using Cauchy's Integral Formula on a simple closed contour

Hello I'm trying to evaluate the following two integrals where C is the unit circle centered at origin, but I encounter the same problem in both of them and can't think of what to do. $$1)\ \oint_{C} ...
0
votes
2answers
54 views

Prove that $|R(z)| \leq \frac{e-1}{(n+1)!}$ if $|z| \leq 1$ Complex Variables

Let $R(z)$ be the remainder after $n$ terms in the power series of $e^z$. That is $$R(z) = e^z - \sum_{k=1}^{n}\frac{z^k}{k!}=\sum_{k=n+1}^{\infty}\frac{z^k}{k!}$$ Prove that $|R(z)| \leq ...
0
votes
0answers
27 views

proof of an integration identity

I could not figure out why the following identity holds: $$ \begin{align} & \int^1_0 dt \int_{\{|z|=1-\varepsilon\}} \, dz\frac{\phi(z)-1}{\phi_t(z)} \int_{\{|s|=1\}} \, ds \frac{\log ...
0
votes
2answers
38 views

Under which conditions is $\int_\Gamma f=0$

Let $\Gamma$ be a contour. $V\subseteq \mathbb C$ and $f:V\rightarrow \mathbb C$. What are the conditions on: $\Gamma$ $f$ Open $U\subseteq V$ such that $\int_\Gamma f=0$ ...
1
vote
2answers
33 views

Indefinite integral

I want to integrate $\frac{1}{(1+x^4)}$ from zero to infinity, set $z_0,z_1,z_2$ and $z_3$ to be the roots of: $1+x^4$ Using Cauchy integral formula, on which path I should integrate?
0
votes
1answer
21 views

Find $\oint_\Gamma \frac{\cos z}{z(z^2+8)}dz$ where $\Gamma$ is a positevly oriented unit circle.

Find $$\oint_\Gamma \frac{\cos z}{z(z^2+8)}dz$$ where $\Gamma$ is a positevly oriented unit circle. My attempt: $$\oint_\Gamma \frac{\cos z}{z(z^2+8)}dz=\oint_\Gamma \frac{\cos z}{z(z-\sqrt8 ...
0
votes
0answers
25 views

For a real number $r>0$ for the function $\cot(z)$ has a Laurent series where 0<|z|<r. What is the largest possible r?

Okay so I know that $\cot(z)$ expands as: $$ \cot(z) = \frac{1}{z} - \frac{z}{3} - \frac{z^3}{45} - \ldots$$ But I'm unsure how to find the largest $r$. I'm assuming this is the point where it ...
2
votes
1answer
123 views

Fourier transform with branchcuts

I would like to compute this kind of Fourier transform $$I=\frac{1}{2\pi}\int_{-\infty}^{\infty}dw\; e^{-i w t}\frac{1}{\sqrt{1-w^2}(w^2+\epsilon^2)}$$ which has a branchcut and $\epsilon\in{\rm ...
1
vote
2answers
17 views

Piecewise continuous contours with discontinuity only at end points

Let $w(t)=u(t)+iv(t)$ where $a \leq t \leq b$ be a complex valued function on real variable $t$. For integrating $w(t)$ from $a$ to $b$ we require that $u(t) $ and $v(t)$ must be piecewise ...
2
votes
1answer
32 views

Computing the residue of $\frac{\cot(\pi z)}{z^2}$ at pole $z=0$

To find the residue I used the residue theorem that states: $$Res(f,z_0)=\frac{1}{(m-1)!}\lim_{z\to0}\frac{\mathrm{d}^{m-1}}{\mathrm{d}z^{m-1}}(z-z_0)^m f(z)$$ where $m$ is the order Computing the ...
2
votes
1answer
71 views

Evaluating the integral $\int_{-\infty}^{\infty} \frac{(1-ix)^{n-s_1}}{(1+ix)^{n+s_2}} dx$.

How can one show that for $s_1,s_2 \in \mathbb{C}$ $$ \begin{aligned} \int_{-\infty}^{\infty} \frac{(1-ix)^{n-s_1}}{(1+ix)^{n+s_2}} dx = & 2\sin(\pi(n+s_2))2^{1-s_1-s_2} ...
2
votes
1answer
87 views

Fourier transform of Gaussian divided by a polynomial

I am trying to compute the following Fourier transform $$\int_{-\infty}^\infty\text{d}x\,e^{i k x}e^{-x^2/a^2}\frac{1}{[x-(d-i\epsilon)]^3}$$ where $d\in\mathbb{R}$, and $\epsilon$ is a small, real, ...
2
votes
0answers
105 views

Fourier transform of a Gaussian times a rational function

I am trying to compute the following Fourier transform $$\int_{-\infty}^\infty\text{d}x\,e^{i k x}e^{-x^2/a^2}\frac{P(x)}{Q(x)}$$ where $\text{deg}P(x)+1\leq\text{deg}Q(x)$, and the roots of $Q(x)$ ...
1
vote
2answers
52 views

Prove $\int_{-\infty}^{\infty} \frac{dx}{(1+x^2)^{n+1}} = \frac{(1)(3)(5)…(2n-1)}{(2)(4)(6)…(2n)} \pi \ \ \ \forall n \in \mathbb{N}$

My attempt starts with a contour integral in the half disk, I let the radius -> infinity and so the contour integral \begin{equation} \int_{\gamma} \frac{dz}{(1+z^2)^{n+1}} = 2 \pi i \ res_{z_0 = i} ...
1
vote
1answer
30 views

Confusion over complex integral along a path

Compute $$I:=\int_C\frac{z^9}{5}dz,$$ where $C$ is the curve $z(t)=\sin t+i\sin10t$, $0\leq t\leq\pi/2$. Would the answer be: $I=\int^{z(\pi/2)}_{z(0)}\frac{z^9}{5}dz$ where $z(\pi/2)=1$ and ...
1
vote
2answers
35 views

Question involving the Cauchy-Goursat Theorem

Show $\int_C \frac{dz}{(z-z_0)^n} = 2\pi i$ when $n = 1$ and 0 when $n \gt 1$ I'm not entirely sure how to approach this one. Any help would be much appreciated.
2
votes
1answer
51 views

Modulus of a Complex Logarithm

I'm currently self-studying complex analysis and I'd like to analytically show that $$\lim\limits_{R\to\infty}\int_{\gamma_R} \frac{\ln\left(z+i\right)}{z^2+1}\ \mathrm dz=0$$ Where $$ ...
1
vote
0answers
36 views

Let $f:U\rightarrow \mathbb{C}$ holomorphic no constant, then $\mbox{Re}f+\mbox{Im}f$ has no minimum in $U$.

Let $U\in\mathbb{C}$ be an open and connected set and let $f:U\rightarrow \mathbb{C}$ holomorphic. Suppose that $f$ is not constant. Show that $\mbox{Re}f+\mbox{Im}f$ has no minimum in $U$. Remark: I ...
0
votes
2answers
36 views

Complex analysis: Evaluate the integral

Evaluate $\int(z^2-z+2) dz$ from $i$ to $1$ along the contour $C$ given in the figure. The figure shown is the line $y=1-x^2$ from i to 1. I'm having trouble parameterizing this curve. If ...
0
votes
2answers
45 views

Evaluate using complex integration: $\int_{-\infty}^\infty \frac{dx}{(x^2+1)(x^2+9)}$

Evaluate $$\int_{-\infty}^\infty \frac{dx}{(x^2+1)(x^2+9)}$$ Firsly I found the residues of this function: $$Res(i)=-i/16$$ $$Res(-i)=i/16$$ $$Res(3i)=i/48$$ $$Res(-3i)=-i/48$$ I then closed ...
3
votes
1answer
84 views

Use Cauchy Inequalities to find an upper bound for $|f^4(i)|$ and $|f^4(0)|$

Let's suppose that $f$ is differentiable on a disk $B_{10}(r)$, and $|f(z)| \leq 54$ for $z$ on the circle $|z-i| = 3$. My goal is to use Cauchy Inequalities to find an upper bound for $|f^4(i)|$ and ...
1
vote
3answers
60 views

Prove that $\int_{0}^{\pi}e^{\cos(\theta)}\cos(\sin(\theta))\,d\theta = \pi$

I have to prove that $$\int_{0}^{\pi}e^{\cos(\theta)}\cos(\sin(\theta))\,d\theta = \pi$$ by considering $$\int_{\gamma} \frac{e^z}{z}$$ where $\gamma(t) = e^{2{\pi}it}, 0 \leq t \leq 1$. I have ...
2
votes
2answers
53 views

For what simple piecewise smooth loops is this integral zero?

I'm trying to solve the following problem: For what simple piecewise smooth loops $\gamma$ does the following equation hold: $$\int\limits_\gamma \frac{1}{z^2 + z + 1} \, \mathrm{d}z= 0$$ ...
0
votes
1answer
24 views

Complex integration

f(θ)=(1/1+r^2 -2rcosθ) , I have calculated this integral: Integral(θ= 0 ->2pi) f(θ) dθ = (2pi/r^2 -1) ,How to find: Integral(θ= 0 ->2pi) (cos(nθ)*f(θ)) dθ
0
votes
0answers
32 views

Prove f is zero in the interior of a closed curve

If $f$ is an analytic function on a simple closed curve $(γ)$ and its interior, and if $f=0$ on $(γ)$ How to prove that $f=0$ on the interior of the curve $(γ)?$
0
votes
2answers
23 views

Why $|dz| = -i \rho \frac{dz}{z}$?

I am solving exercise $3$ p. 120 in Ahlfors' C.A. Find $$\int_{|z| = \rho} \frac{|dz|}{|z - a|^2}$$ Hint is given: $$|dz| = -i \rho \frac{dz}{z}$$ I don't understand why this is true. Please ...
2
votes
1answer
39 views

Hermite Polynomials: Rodrigues to Integral Representation

I would like to go from this representation of the Hermite polynomials: $$H_n(z)=(-1)^ne^{z^2}\frac{d^n}{dz^n}e^{-z^2} \tag{1}$$ to this representation ...
0
votes
0answers
21 views

Regularity of complex function in region $D$

We say $f(z)$ is regular in $D$ if for every point $z\in D$, $f$ is differentiable. The question comes with regards to Cauchy formula and Newton-Leibniz formula for $$\int_\gamma ...
2
votes
1answer
42 views

If $f(z)$ analytic on $E(L)$, and $\lim_{z \to \infty}f(z)=A$, prove that $\frac{1}{2\pi i} \int_{L}\frac{f(\zeta)}{\zeta-z}dz = A$ or $-f(z)+A$

I am working on the following problem: Let $L$ be a closed, rectifiable simple curve, traversed counterclockwise. Let $f(z)$ be a differentiable function on a domain $G$ where $L \cup E(L)$ ...
8
votes
2answers
114 views

closed form for $I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$

$$I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$$ for $n=1$ I tried to use $\arctan x=u$ and by notice that $$\frac{1+\tan u}{1-\tan u}=\cot\left ( ...
1
vote
1answer
38 views

Calculating complex integral via definition

Consider $$\int_\gamma \frac{1}{4z+iz^2}\mbox{d}z $$ where $\gamma$ is the boundary of the triangle with vertices $-i,2+i,-2+i$. The assignment is to solve it using the definition which is given as ...
2
votes
1answer
46 views

Determine all the biholomorphic functions $\mathbb{C}\rightarrow \mathbb{C}$.

Determine all the biholomorphic functions $\mathbb{C}\rightarrow \mathbb{C}$. My attempt: First, we show that $z_{0}=0$ is not essential singularity of $g(z)=f\left(\frac{1}{z}\right)$, indeed, if ...
0
votes
1answer
31 views

Converting certain complex exponentials to trigonometric functions

The original question is: Show that $$f(x)=\sum_{k=-\infty}^\infty c_k e^{-i kx}=\frac{2\sinh(\pi)}{\pi} \sum_{k=1}^\infty \frac{(-1)^{k-1}k}{1+k^2}\sin(kx)$$ where $\displaystyle ...
3
votes
1answer
76 views

Use of residues to find I=$\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$

I'm working on the problem $$I=\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$$ I found 4 singularities and i would like to use the singularities in the 1st and 2nd quadrants to solve this integral; i.e. ...
0
votes
2answers
14 views

$\int_{[1+iR, 1+2iR]}\frac{e^{z^2}}{z^2}dz \to 0$ as $R \to \infty$

Please help me with this problem, $\int_{[1+iR, 1+2iR]}\frac{e^{z^2}}{z^2}dz \to 0$ as $R \to \infty$ My attempt : $r(t)= 1+ itR$, $1\leq t\leq 2$ Then the integral becomes $\int_1^2\frac{e^{(1+ ...
5
votes
1answer
58 views

Use Cauchy's Theorem to show that if $\int_{0}^{\infty}f(x)dx$ exists, then so does $\int_{L}f(z)dz$

Suppose that $f(z)$ is analytic at every point of the closed domain $0 \leq arg z \leq \alpha$ $(0 \leq \alpha \leq 2 \pi)$, and that $\lim_{z \to \infty}z f(z) = 0$. I need to prove that if the ...
1
vote
0answers
37 views

If $J(L)=\int_{L}\frac{dz}{p(z)}$, where $p(z)$ is a polynomial with $n$ distinct roots, how many values can $J$ take on?

My question is related to this one, but it is not a duplicate, because I am not allowed to use Residues or Cauchy's Integral Formula to solve it. The only tools I have at my disposal are Cauchy's ...
5
votes
2answers
52 views

Evaluate $\displaystyle \int_{|z-i|=R}\frac{z^{4}+z^{2}+1}{z(z^{2}+1)}dz$ as a function of $R>0$

I need to evaluate the integral $\displaystyle \int_{|z-i|=R}\frac{z^{4}+z^{2}+1}{z(z^{2}+1)} dz$ as a function of $R>0$. I may omit values of $R$ for which the denominator turns to $0$. Now, ...
0
votes
2answers
42 views

Why is the orthogonality of complex functions defined with the integration between its complex conjugate?

As stated in the title, I am not quite understand the orthogonality of complex functions. For example, for the following function family: \begin{equation} \phi_k(x)=e^{ikx}, \end{equation} its ...
2
votes
2answers
107 views

Without Cauchy Integral Formula or Residues: $\int_{L}\frac{dz}{z^{2}+1}$ [duplicate]

I am currently working on the following: Prove that $\displaystyle \int_{L}\frac{dz}{z^{2}+1} = 0$ if $L$ is any closed rectifiable simple curve on the outside of the unit disc, i.e., $L$ is ...
2
votes
2answers
55 views

Question about an integral

Where can I find the full derivation or the proof of the following integral? $$\lim_{x \rightarrow +\infty} \int_0^{x} dx'e^{-i(k-k_0)x'} = \pi\delta(k-k_0) - P(\frac{i}{k-k_0})$$, where $P$ stands ...
6
votes
1answer
82 views

Determine the Integral $\int_{-\infty}^{\infty} e^{-x^2} \cos(2bx)dx$

Please do not mark this question as a duplicate. I have to solve this with a different method that I don't believe has been discussed about this particular question (at least to my knowledge). I am ...
0
votes
0answers
75 views

How to compute the following integral

I have to compute the following integral directly. I was trying to simplify with some change of variable but it becomes always more complicated. Is there a trick to compute it easily? ...
0
votes
1answer
40 views

Integration of $\log z$ in complex analysis

Let $u(w)$ be a compact support, real valued, smooth function on $\mathbb C$(hence also on $\mathbb R^2$), can we define $\eta(z) = \int_{\mathbb {R^2}} \log(|z-w|) \, u(w) \, \mathrm{d}x \mathrm{d}y ...
2
votes
1answer
99 views

Variation of argument of a complex function

Variation of Argument : Definition( Collect from my book ) : Let $f$ be analytic inside and on a sinple closed contour $C$ except possibly for poles inside $C$ and $f(z)\not=0$ on $C$. As $z$ ...
0
votes
1answer
50 views

Prove that the following function has a branch cut

I am given with a function $$\zeta(z)=\int_{-\infty}^\infty\mathrm{d}x \frac{f(x)}{z-x}.$$ Any idea to prove that $\zeta(z)$ is discontinuous across real axis for $f(x)\neq 0$?
2
votes
1answer
30 views

Show that $f'(a)=\frac{1}{2\pi}\int_0^{2\pi}\mathrm{e}^{-\mathrm{i}\theta}f(a+\mathrm{e}^{\mathrm{i}\theta})\mathrm{d}\theta$

Here is a picture of the problem (sorry I have trouble typing the symbols). Suppose f is holomorphic inside of a simple closed curve gamma. Show that ...
0
votes
0answers
69 views

Surface integral of complex function using residues

I am stuck with a math problem that I thought should be straightforward. Maybe I'm missing something here and you can help me. The key idea is that I have to integrate this function over a volume: ...
1
vote
1answer
26 views

Determining the integral of $f(z)\, dz$ and expressing the answer in terms of residue.

Suppose that $f : \mathbb{C} \setminus \{0, ±i\} → \mathbb{C}$ is the rational function $$f(z) = {a_{−3}\over z^3} −{a_{−2}\over z^2}+{a_{−1}\over z}+ a_0 + za_1 +{z\over (z^2+1)}.$$ Let $Γ$ denote ...
7
votes
2answers
109 views

Proof using Cauchy Integral Theorem

Suppose that I have the following integral: $$ \int_L \frac {dz}{z^2+1} $$ I need to show that this is equal to $0$ if $L$ is any closed rectifiable simple curve in the outside of the closed unit ...