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

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5
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1answer
120 views

How to calculate this complex integral $\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$? (Please Help)

I want to carry out the following integration $$\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$$ which is trivial if calculated numerically with any value for b. But I really need to get an ...
2
votes
1answer
46 views

Can use residue theorem for this integral

I need to compute $$I=\int_C \dfrac{e^{\sqrt{1+u}}\cdot\sqrt[4]{1+u}}{\sqrt{u}} \,\mathrm {d}u$$ where $C$ is the unit circle. I am confused about whether I can use the residue theorem to compute it? ...
2
votes
1answer
59 views

Integration of complex function

how do we do start doing the following integral? $$\int_{-\infty}^{\infty}e ^{-\pi(x^2a^2+2iux)}\,dx$$ Any help will be appreciated NOTE: $i$ is the square root of $-1$.
1
vote
1answer
36 views

Complex Integration and deduce that function is constant

Let $f$ be an entire function, $z_{1}$, $z_{2}$ $\in$ $C$, with $z_{1} \neq z_{2}$ and $R>\max{(|z_{1}|,|z_{2}|)}$. Prove that $$2\pi i\dfrac{f(z_{1})-f(z_{2})}{z_{1}-z_{2}} = ...
1
vote
1answer
27 views

Fourier series representation of $\sin^4 x$

I tried solving for fourier coefficients of Fourier series for the multiples of fundamental frequency $\omega_0=2$. So $F_n=\int_0^{\pi} \sin^4 x \, e^{-i2nx} dx$. And my calculator says answer should ...
1
vote
1answer
29 views

Complex integral with different contours

If I have a complex integral to solve using the Cauchy Integral formula with the same point but with different contours, in which the point used is inside both contours, is the result the same? Say ...
1
vote
1answer
42 views

Is Cauchy's integral theorem affected by integral direction?

hello,everyone,I hava a exam question aboat the integraion~ as shown below I know 1/(Z^2-1)=1/2(1/[z-1]+1/[z+1]) the integrantion around 1 should be 2*pi*i, but i am confused if the integrantion ...
1
vote
1answer
41 views

Integral of $e^{\overline{z}}$

So I'm suppose to integrate $e^{\overline{z}}$ along the unit circle where $z(t)=e^{it}$ and from t goes from 0 to $2\pi$. This is the work I've done so far $\int e^{\overline{z}}dz = ...
1
vote
1answer
90 views

How can I use Cauchy formula to this Integral?

$$∫ \frac{5z^2 - 3z + 2}{(z-1)^3} dz$$ and the contour is any closed simple curve involving z=1 (sorry, I forgot to write this information) Need to solve it using Cauchy Integral formula Can anyone ...
0
votes
1answer
31 views

inequality in absolute value with exponential

Can you help me with this problem please ?? If $r>0$ . Show that $\left |{\displaystyle\int_{\gamma}e^{iz^{2}}dz}\right |\leq{\displaystyle\frac{\pi(1-e^{-r^{2}})}{4r}}$ where ...
7
votes
0answers
65 views

closed form for $I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx$

solve $$I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) dx:0<n<2$$ I tried for $n=1$ : $$I(v)=\int_{0}^{\infty}\frac{x}{x^2+u^2}\tanh(vx) dx$$ ...
3
votes
0answers
1k views

Integrate complex conjugate

I'm doing some exercises on complex integration, and stumbled over this: $$\int_\gamma \bar{z} dz,$$ Where $\gamma$ is any closed $C^1$ curve which is the boundary of a bounded, connected, open $U ...
2
votes
0answers
25 views

Contour Integral of sin(z)/(z^2-z)

Find the integral $\int_{\lambda}\frac{\sin(z)}{z(z-1)}$ where $\lambda(t) = 10e^{it},t\in[0,2\pi]$ We notice that there are poles at $z = 0$ and $z=1$. So we can use residue theorem but I am ...
2
votes
0answers
49 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
2
votes
0answers
33 views

$ g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm {d}b$

Do you have any idea how to compute this function? $$ \displaystyle g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm ...
2
votes
0answers
130 views

Show Smoothness by Morera

I'm trying to show smoothness on $(0,\infty)(\Re)$ of the following function: $$ f(t,x)=\sum_{n=-\infty}^\infty e^{-\large \frac{(x-2\pi n)^2}{2t}}\frac{1}{\sqrt{2\pi t}} $$ The function is ...
1
vote
0answers
29 views

Complex integration, limits, arctan

$\left.\frac12i\;\text{Log}\frac{1-(-i+e^{it})}{1+(-1+e^{it})}\right|_0^{2\pi}=\frac12i\left(\log\left|\frac{i}{i}\right|+i\arg 1-\log|1|-i\arg1+2\pi ik\right)$ could someone explain how this is ...
1
vote
0answers
27 views

If f is analytic in the unit disk and $|f(z)|\leq \frac{1}{1-|z|}$ then $|f'(0)|\leq 4$

I need to prove that if $f$ is analytic in the unit disk and $|f(z)|\leq \frac{1}{1-|z|}$ for z all $z\in D_1(0)$ then $|f'(0)|\leq 4$. This is my proof and I need to verify this. Let $n\in ...
1
vote
0answers
30 views

How to integrate $\int\limits_C{\frac{sin\pi z}{(z^2-1)^2}}dz$, where $C: |z-1|=1$ using Cauchy's formula?

How can evaluate $$\int\limits_C{\frac{sin\pi z}{(z^2-1)^2}}dz$$, where $$C: |z-1|=1$$ by using Cauchy's formula. I have to use Cauchy's formula. Cauchy's formula $$f(z_0)=\frac{1}{2\pi ...
1
vote
0answers
21 views

Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
1
vote
0answers
22 views

Explicit computation of integrals along loops

I am learning a bit about integration on one-dimensional complex tori. It's exciting stuff, but I have some trouble making things a bit explicit. Let's consider the elliptic curve $E = \mathbb ...
1
vote
0answers
28 views

Find natural number to satisfy inequality given with integral

Find the number $N\in \mathbb{N} $ for which the following inequality is true, if $\sigma(t)=|z|=1 $, with $0\leq t\leq \pi/2$ and $\left | \int_{1}^{i} e^{z^{-1}} \right | \leq N \frac{\pi}{2}$. ...
1
vote
0answers
64 views

Calculating this integral: $\int_{\partial_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}z^z}{(z+4)^{42}}dz$

Please take a look at $$\int_{\partial B_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}ze^z}{(z+4)^{42}}dz$$ At a first glance, this looks like a case for Cauchy's differentiation formula, which ...
1
vote
0answers
24 views

Evaluate the bound of the following complex integral

Let $q$ be a complex analytic function such that $\left|q\right|\neq0$ everywhere in the domain, except at the boundaries ($\left|q\right|\rightarrow0$ as $z\rightarrow\pm\infty$). Is it possible ...
1
vote
0answers
32 views

Second Derivative of Holomorphic function According to Cauchy's

I am asked to prove that, supposing f is holomorphic on the region G, w $\in$ G and $\gamma$ is a positively oriented, simple, closed, smooth, G-contractiblecurve such that w is inside $\gamma$ , we ...
1
vote
0answers
38 views

Integration containing a complex number

Folks, Can I treat the complex number in the following integral: $$\frac1{2\pi}\int\frac1{(1+jw)^2}dw$$ as a constant and move it outside of the integral, like this: ...
1
vote
0answers
132 views

Evaluate the following integrals/ Cauchy integral theorem

So I have two questions. 1) Evaluate $ \oint_{|z|=1} \dfrac{cos(\pi z^2)}{(z-2)(z-4)^3} dz$ and 2) Evaluate $ \oint_{|z|=6} \dfrac{cos(\pi z^2)}{(z-2)(z-4)^3} dz$. Now I know the integrand is ...
1
vote
0answers
119 views

Complex integration from zero to infinity at different directions

The problem is computing the integral expression $$f(t)=\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$$ where z is a complex variable with $Re(z)\ge 0$. Is it correct to substitute $w=\lambda z$ ...
1
vote
0answers
132 views

Analytical Continuation

I have problem understanding the analytical continuation of the following complex integral. I even have the solution but can not figure out the intermediate steps. I would appreciate it if anyone ...
1
vote
0answers
51 views

Integral of the determinant of the Jacobian in complex variables?

Let $f$ be a function that maps a disc $D$ onto a unit square $A$ such that $f(D) = A$. $f$ is holomorphic and one-to-one. Why does $\int_D|f'(x+iy)|^2 \, dx \, dy = 1$?
1
vote
0answers
257 views

Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
1
vote
0answers
214 views

Residue of $\mathrm{exp} (z+1/z)/((z-z_1)(z-z_2))$ at z=0

The ultimate aim is to solve the following integral: \begin{equation} \label{eq:Icos1} \begin{aligned} I = \int\limits_{0}^{2\pi} \frac{\mathrm{exp}\left(c \cos(\varphi)\right)\mathrm{d}\varphi}{a - ...
0
votes
0answers
26 views

Expressing principal value of integral as real/imaginary

How is it that we can express $$ \mathrm{p.v.}\int_{-\infty} ^{\infty} \frac{\cos 3x}{x^2+4}=\Re \ \mathrm{p.v.}\int_{-\infty} ^{\infty} \frac{e^{3xi}}{x^2+4} $$ while we cannot for $$ ...
0
votes
0answers
48 views

How to show that a given domain is simply connected?

I was studying simply and multiply connected domains in complex integration. I know that the domain $\{z: 1<|z|<2\}$ is multiply connected as it can't be squeezed to a point without going ...
0
votes
0answers
23 views

Could anyone help me to solve this integral question?

Could anyone help me to solve this integral question ? $$ \ \int_a^b t^{k-z-1}(1+mt^{-z})^{-(n+1)}dt $$
0
votes
0answers
56 views

Calculate $\int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx$ using principal branch

I would like to calculate the following integral $$ I = \int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx $$ using contour integration but using principal branch of the function, i.e. ...
0
votes
0answers
23 views

Integration of a complex integration

Let $C$ be the contour $|Z|=2$ oriented in the anti-clockwise direction.What is the value of the integral $\oint_{C}{ze^{\cfrac{3}{z}}}$$dz$ ? I don't know how to start. Please tell me which formula ...
0
votes
0answers
27 views

Plemelj like relation

How to prove the identity: $\frac{1}{(x+i\epsilon)^{n+1}} = P \frac{1}{x^{n+1}} - i \pi \frac{(-1)^n}{n!} \delta^{(n)}(x)$ that holds when integrating ($\epsilon$ is infinitesimal, x, $\epsilon$ are ...
0
votes
0answers
20 views

Sum of Lomax random variables

Suppose $X_1,X_2,\cdots X_n$ are $n$ i.i.d Lomax random variables with pdf $f(x)=\frac{m}{(1+x)^{m+1}},x\geq 0,m\in \mathbb N$. I need to determine the pdf (or cdf) of the sum $S_n=\sum_{i=1}^{n}X_i$. ...
0
votes
0answers
11 views

Contour Integral squared function

$\gamma$ is a contour that goes from $-i$ to $-1$ and is contained in the third quadrant. Calculate $$\int_{\gamma}{z^{\frac{1}{2}}}dz$$ It is obvious that the primitive of the function is $\dfrac{2 ...
0
votes
0answers
46 views

Complex Integration Over an Ellipse

How do we evaluate $\int1/\sqrt{1-z^2}dz$ over the ellipse with the standard form with $a^2$$-$$b^2$=$1$$?$ I was trying to use the Cauchy's Integral Formula and the fact that a circle is homotopic to ...
0
votes
0answers
23 views

Change of variable in complex integral

When I want to evaluate the following integral $$\int_\Gamma e^{-z^2}dz~~~~~~~~~~~\Gamma:|z|=R,0\leq\arg{z}<\frac{\pi}{4}$$ so I subtitude, letting(choose a single-valued analytic branch) ...
0
votes
0answers
35 views

Boundary line integral

I was trying to do this integral $$ \oint_{\left\vert\,z - 2\,\right\vert\ =\ 2}z^{4}\sin\left(\, z\,\right)\,{\rm d}z $$ by the definition of line integral $\displaystyle\int_{a}^{b}{\rm ...
0
votes
0answers
43 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 ...
0
votes
0answers
96 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
0
votes
0answers
41 views

Solving an ordinary differential equation with two functions

Please I am trying to solve this differential equation but I do not seem to get the correct answer. Please help thank you. My equation is; $\frac{dc(x)}{dx} = 1+ b\frac{y(x)-y(x-1)}{y(x-1)}$ where b ...
0
votes
0answers
47 views

A question on particular functions in $L^\infty$

Let D be the open unit disk in the complex plane, $\mu$ be the arc length measure, and $f, g\in L^{\infty}(\partial D,\mu)$ satisfying the following equations: $$ \int_{\partial D} ...
0
votes
0answers
32 views

Contour integral going to zero on a limit

I've been asked to prove the following; $$\lim_{R \rightarrow \infty}\int_{C_R} \frac{z^2 + 8z + 42}{(z^2+4)(z^2-4z+5)}dz=0$$ Given that $C_R$ is a circle of radius $R$ centered at $0$. I thought ...
0
votes
0answers
51 views

Liouville's theorem for functions not bounded on an isolated set

On Liouville's theorem. Prove that if $f(z)$ is defined, analytic, bounded in the entire complex plane, except for an isolated set of points, then $f$ must be constant. What happens if said set has a ...
0
votes
0answers
35 views

Integration of a complex function having square rooted denominator

How to evaluate this integral ? $$\int_{-\infty}^{\infty}\frac{1}{(w-ia)\sqrt{w^2-ibw+b^2c}}dw$$ where $a$, $b$ and $c$ are real and greater than zero. Does the residue theorem work on it? Please ...