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1
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3answers
104 views

Calculating the integral expression $\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$ for complex-valued z

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 $Re(z)> 0$ and t is a real variable. Is it correct to ...
5
votes
1answer
111 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
54 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
37 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
68 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
28 views

Lioville's criteria and Integration of sin(z)/z .

Lioville Criterion: $\int fe^g$ is elementary iff there is rational function $q$ in $\mathbb{C}(z)$ such that $$f=q'+qg'$$ where $f\neq0$ and $g$ is a non constant function in $\mathbb{C}(z)$. Now ...
0
votes
1answer
45 views

Conditions for complex integrability

I was wondering if the Lebesgue conditions for Riemann integrability also hold for a complex path integral, and what that would mean exactly. I am assuming that $f(z)$ would have to be bounded and ...
0
votes
1answer
80 views

A taylor series for an integral with a singularity

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function with a single root we call $y_*$. Then define \begin{equation} F_{\delta}:=\int^{y_*+ \delta/2}_{y_*- \delta/2} 1/f(y)dy ...
2
votes
0answers
42 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
385 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
32 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
125 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
31 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
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0answers
72 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
85 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
90 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
39 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
210 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
197 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
33 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
23 views

Evaluate $\int f(z)\,dz$ with $f(z) = x^2 - iy^2$, where $z = x + iy$, and the curve $C$ is given by $C(t) = t - it^2, 0<t<1$

I began by assigning the $\Re z$ to be $x = t$ and the $\Im z = y = -it^2$. Then I computed $z'(t) = 1 - 2ti$. Then $f(z(t)) = t^2 - it^4$. Then I took $$ f(z(t)) z'(t) = (t^2 - it^4)(1-2ti) = (t^2 ...
0
votes
0answers
28 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 ...
0
votes
0answers
31 views

Relation of an integral over the entire real line to the same but with the integrand shifted by an imaginary amount

I would like to relate the following two integrals: \begin{align} I_1 &= \int_{-\infty}^\infty f(x) dx .\\ I_2 &= \int_{-\infty}^\infty f(x - i X) dx \text{ with } X \text{ real.} \end{align} ...
0
votes
0answers
119 views

complex sum-integration

Let $a_n \in \mathbb{R}$. Show that, $$0\le \sum_0^N\sum_0^N\frac{a_na_m}{n+m+\frac{1}{2}}\le \pi \sum_0^Na_n^2.$$ Hint: Integrate on the semicircle with radius R, the sum ...
0
votes
0answers
32 views

Complex Integration using polar coordinates

Consider the complex variable $y=re^{j\phi}$ with $r\in(0,\infty)$, $\phi \in (-\pi,\pi)$, and the complex integral $$ I=\int\limits_\mathbb{C} {f(y)\log(f(y))dy} $$ Does the following ...
0
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0answers
22 views

Integration of error function and exponenial with none trivial integration limit

I would like to know the following integration $$\int_b^\infty \operatorname {erfc(x)}e^{x^2+iax}$$ which seems integrable as the integrand goes to $\frac{e^{ix}}{x}$ for large x. All reference ...
0
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0answers
35 views

Complex Integration with two variables

Let $ z_1, z_2 \in \mathbb{C}^+ \equiv \{ \forall z \in \mathbb{C}^+, \Im(z)>0 \} $. Also define $ m_1, m_2 $ such that for $i=1,2$ the following is true $$ z_i=-\dfrac{1}{m_i} + ...
-1
votes
0answers
34 views

proving cauchy integral formula using integration by parts

I have found several proofs of Cauchy's generalized Integral formula, but I am looking to prove the first derivative case by using the fact that the first derivitave is analytic to state it as a ...