0
votes
1answer
16 views

calculate integral $ln(z) $

Hi everyone, Pleaze is it possible to calculate $\int_{C(0,1)}ln(z) dz$ using the residue theorem ??? Thank you for your help. <\br>
0
votes
2answers
34 views

Complex integration along a curve

I have to calculate this integral: $$ \int_C e^z\,dz $$ where $C$ is the circle $|z - jπ/2| = π/2$ from the point $z = 0$ to the point $z = jπ$. I know how to calculate these with circles which ...
0
votes
1answer
35 views

Evaluating the integral of $1+z+1/\tan z$ over a circle

I am a beginner and I want to learn how to solve these kind of integrals: $$\int_{|z|= \pi/4}\left(1+z+\frac{1}{\tan z}\right)\,dz$$ So should I divide it in three integrals, calculate each integral ...
-1
votes
0answers
18 views

Integral vanishes as $r \rightarrow \infty$, exponent

I'm integrating $e^{-x^2}$ over the boundary of a triangle and in order to finish my solution I need to show that $$\int_0^r (e^{y^2-r^2-2iry})i dy$$ vanishes as $r$ approaches infinity. Could you ...
0
votes
1answer
18 views

Solve the integral with complex number and floor function.

let $z\in\mathbb{C},\,0<\left|z\right|<1$.I would like to calculate the integral $$\int_{1}^{\infty}\frac{\left(1-z\right)^{\left\lfloor t\right\rfloor }}{t^{2}}dt$$ where ${\left\lfloor ...
2
votes
1answer
27 views

Half-Fourier transform, relation to Delta function

so the Fourier transform of the Kronecker Delta function is (up to sign conventions / normalisation) $$\int_{-\infty}^\infty dt\; e^{i t \omega} = \delta(\omega).$$ Can one say anything about the ...
0
votes
0answers
16 views

asymptotic esimation of a complex integral

I am searching for a general method to evaluate asymptotically this kind of integral $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(q,\omega)\exp[-\mathrm{i}kr]\exp[-\mathrm{i}\omega ...
2
votes
1answer
41 views

Line integral along Circle

Can you please give me a hint so that I can get started with the following integral? $$\oint_{C(R)} \frac{\sin(\pi/z)}{(1+z)^2} dz$$ where $C(R)$ is a circle with radius $R$ and origin $0$. ...
1
vote
0answers
29 views

How to prove that the integration contour of one integral is the subset of the other integral? [on hold]

I have the two integrals which have the same non negative integrand. For example the following two integrals, $\int_{-l}^{l}f(x)dx$ ....... (1) and $\int_{a}^{b}f(x)dx$ ....... (2) What we ...
0
votes
1answer
28 views

Compute $\int_\gamma\overline{\zeta} \, d\zeta$ using Cauchy’s Integral Formula

Let $\gamma$ be the circle of radius $1$ and centre $0$, equipped with the counterclockwise orientation. Compute $$\int_\gamma\overline{\zeta} \, d\zeta$$ using Cauchy’s integral formula. Any hints ...
1
vote
2answers
60 views

Evaluate $\int_{\partial C} \frac{dz}{(z-a)(z-b)}$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are not on $\partial C$)

In discussing the possible outcomes of the integral $$\int_{\partial C} \frac{dz}{(z-a)(z-b)}$$ where $\partial C$ is the boundary of a rectangle ($a$ and $b$ are complex and not on $\partial C$), ...
1
vote
2answers
64 views

Evaluating $\int_0^\infty e^{-x}\cos(x)dx$

By integrating over the contour around an appropriate sector, how does one solve $$\int_0^\infty e^{-x}\cos(x)dx$$
1
vote
1answer
10 views

Complex integral, absolute value of integrand

I want to integrate $f(z)=\frac{1-\mathrm{e}^{\mathrm{i}z}}{z^2}$ over the indented semicircle in the upper half-plane positioned on the $x$-axis as pictured below. The book (Complex Analysis by ...
2
votes
1answer
86 views

Determine the following integral (very difficult)

So we let $\sqrt{z}$ be the principal value square root of $z$ (i.e. with $\sqrt{1} = 1$ and branch cut along the negative real axis), also let $a \in \mathbb{R}^+$. Determine the following integral: ...
0
votes
0answers
27 views

Using Contour Integration to Integrate $\frac {x^{-z}}{1+x}$

I would like to evaluate the following integral using complex integration: $$\int_0^{\infty}\frac{x^{-z}}{(1+x)}dx$$ where $z \notin \mathbb Z$. I'm given that the answer is $\frac{\pi}{\sin(\pi ...
2
votes
1answer
41 views

Integration of $1/|z-z_0|^2$ over a circle in the complex plane

I am trying to prove the following problem $$\frac{1}{2\pi R}\int_{\delta B_R(0)}\frac{|dz|}{|z-z_0|^2}=\frac{1}{|R^2-|z_0|^2|}$$ where $\delta B_R(0)$ is the boundary of a circle with centre (0,0) ...
1
vote
0answers
28 views

How are Fox-H functions useful in math?

The Fox H-function, as far as I know, is the most general families of functions - encompassing an even larger family of functions than the already very general Meijer G-function. While I've known ...
0
votes
0answers
30 views

Proving inequality with complex Riemann-Stieltjes Integral

I am trying to prove the following proposition (IV.1.17b) from Conway's Functions of One Complex Variable I: Let $\gamma$ be a rectifiable curve and suppose that $f$ is a function continuous on $\{ ...
0
votes
2answers
54 views

What is the relation between two integrals?

Let us suppose that we have two integrals, $I_1$ and $I_2$ with the same non negative integrand. The integration contour of $I_1$ is a subset of the the integration contour for $I_2$. What can we say ...
0
votes
0answers
26 views

Integration contour as points of set.

If B is the subset of A, I wounder do we have two integration contour or one for this? What will happen if we take AUB i.e. A union B, than do we have one integration contour? and what if we take A ...
0
votes
0answers
22 views

How to select the integration contour

In the following two figures which describe sets, How many possible integration contour we have for the figure 1 and how many integration contour we have for figure 2.
1
vote
0answers
25 views

justification of step in complex integration

What is the justification for the step with the red square next to it, how do we change the integrator like this?
2
votes
3answers
63 views

Conplex/real Integration and poles of function

So I am working on the following problem: Let $\Delta $ be the unit disk centered at origin, and $f$ is holomorphic on $\Delta-\{0\}$. If $$\int_\Delta|f|dxdy<\infty$$ show that $f$ has at most a ...
0
votes
0answers
27 views

Inverse of Mellin transform

I would like to invert the following Mellin transform $M(s)$ of a function $f(x)$ defined on $[0,a]$ with $a>0$ (or get the $x\rightarrow 0$ asymptotics): $$ M(s) = \frac{2a^s}{s-2(1-a^s)} $$ We ...
2
votes
1answer
50 views

Switching $\int$ and $\sum$ proof

Been reading through this proof which seems incorrect: Let $f_n$ be continuous on the curve $C$ and $\sum f_n$ converge uniformly on $C$. Then $\sum\int_Cf_n(z)dz=\int_C\sum f_n(z)dz$ PROOF: ...
1
vote
0answers
37 views

Simple complex line integral over a rectangle

What is the easiest way without using residues to calculate: $$\int_{\gamma} {\overline z \over {8 + z}} dz$$ Where $\gamma$ is the rectangle with vertices $\pm 3 \pm i$ in $\Bbb C$ in the clockwise ...
1
vote
0answers
37 views

Integral equation solution

I have an integral equations of the form $ \int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
13
votes
2answers
170 views

Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
0
votes
1answer
33 views

An integral of Wolstenholme:$\int_0^{+\infty}\frac{\sum_1^n A_k\cos{a_k x}}{x}\mathrm {d} x$ where $\sum A_k=0$ and $a_k>0$

The book by Whittaker and Watson says it's equal to $-\sum_{k=1}^n A_k \log {a_k}$, and attibutes it to Wolstenholme. I believe this readily reduces to the simpler case of evaluating $\displaystyle ...
0
votes
0answers
36 views

What is the Riemann surface of the exponential integral?

I have recently encountered a differential equation whose solution has a term \begin{equation} \frac{1}{2}e^{-\frac{1}{2 \varepsilon} e^{i \tau}} \int_{\tau_0}^\tau e^{\frac{1}{2 \varepsilon} e^{i ...
5
votes
4answers
225 views

Wolfram alpha says that $\int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$

Wolfram alpha says that $$ \int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$$ holds. But it has two different values ($\sqrt{i}$). How should I understand this?
0
votes
0answers
50 views

Complex Gaussian Integral - $\int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}}$?

I found some formulas on books, especially the complex gaussian integral formula: $$ \int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}} $$ for $p,c\in\mathbb C$. Then if $p=i=\sqrt{-1}$, the ...
0
votes
0answers
24 views

The integral of a monomial over the complex sphere

Let $\alpha=(\alpha_{1},...,\alpha_{q})\in\mathbb{N}^{q}$ a multi-index. What is the expression for $$\int_{S}z^{\alpha}\,d\sigma(z),$$ where $S$ is the unit sphere of $\mathbb{C}^{q}$ and $\sigma$ ...
0
votes
0answers
52 views

Applying Green's Theorem to a Closed Complex Contour Integral

How would one apply Green's Theorem to the following complex contour integral: $\oint_\gamma $ $\frac{u^{s-1}}{e^{-u}-1)}du$. Where $\gamma$ is the Hankel Contour (counterclockwise) and R is the ...
4
votes
2answers
98 views

$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx$ and $\int_{0}^{\infty} \frac {\ln(x)}{x^2+b^2} dx$

Prove that $$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx = \frac {\pi}{2e}$$ My approach would be $$\lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x)}{1+x^2} dx$$ and evaluate the limits of the sine and ...
10
votes
1answer
229 views

Contour integration with branch points inside the contour.

In my scientific research I ran into an unpleasant situation with specific type of contour integrals. Being more specific I have problems not with integrals themselves (I can use various numeric ...
0
votes
1answer
31 views

The value of the integral of $f(\zeta)/(\zeta-z)$ for a function holomorphic in exterior domain

Suppose that $f$ is a bounded analytic function on the domain $\{z ∈ C : |z| > 1\}$. (a) Prove that $\lim_{z→∞} f(z)$ exists. (b) Let $L$ denote the limit in (a), and let $Γ_R$ denote a circle $|ζ| ...
2
votes
1answer
35 views

integral calculate by complex analysis methods

Calculate using methods from comples analysis. $$ \int_0^{2\pi} \,\sin ^{2n} \phi\, d\phi$$ So this is how I started: $$\sin^{2n} \phi = \left[\frac{e^{i \phi}-e^{-i \phi}}{2i}\right]^{2n} = ...
3
votes
2answers
77 views

And another real integral to be solved by contour integration

I want to solve $$\int_0^\infty\frac{1}{x^3+x^2+x+1}dx$$ and i have really learned a lot already by failing to solve it. I want to solve it using a clever contour. It is possible to do it using ...
2
votes
0answers
118 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch ...
4
votes
2answers
123 views

How to prove $\int^{\pi/2}_0 \log{\cos{x}} \, \mathrm{d}x = \pi/2 \log{1/2}$

ALREADY ANSWERED I was trying to prove the result that the OP of this question is given as a hint. That is to say: imagine that you are not given the hint and you need to evaluate: $$I = ...
0
votes
0answers
75 views

Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx $$ Here's what I have tried. We have to make a ...
0
votes
0answers
66 views

Find a Harmonic conjugate $v(x,y)$ to $u(x,y)$.

Show that $u(x,y) = \frac{y^2}{x^3+y^3}$ in some domain and find the harmonic conjugate $v(x,y)$ to $u(x,y)$.
1
vote
1answer
36 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
5
votes
1answer
117 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) ...
6
votes
2answers
152 views

A strange answer for $\int _{-1}^1 \log x\; dx$

I typed $\int _{-1}^1 \log x\; dx$ on Wolfram Alpha. It is giving the answer to be $-2+i\pi$. Can someone please explain what is happening?
1
vote
1answer
42 views

If a real polynomial of degree $n\gt 1$ has a root of modulus exceeding all others, is that one a real root?

Suppose $a_nx^n+\ldots+a_1x+a_0=0\; (a_n\in \mathbb{R})$ has $n$ distinct roots $r_1,r_2,\ldots, r_n$ (no multiple roots), and if $\exists r_k$ s.t. $\forall r_i\in\{r_1,r_2\cdots r_n\}-\{r_k\}$, ...
3
votes
1answer
49 views

Finding all the possible values of an Integral in the Complex Plane

I am studying Complex Analysis by Lars V Ahlfors. I am unable to solve one of his exercises. It is: Find all possible values of $$\int \frac{dz}{\sqrt{1-z^2}}$$ over a closed curve. I do not have ...
2
votes
0answers
56 views

Can this modified Gaussian integral be calculated analytically?

In my research, I encounter this modified Gaussian integral $$\int_{-\infty}^{\infty}dx\,\frac{x+\sqrt{x^2-bx}}{2\sqrt{x^2-bx}}\exp\left[-a^2(x-x_0)^2+i\left(cx-d\sqrt{x^2-bx}\right)\right],$$ where ...
3
votes
1answer
98 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following two integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...