Complex integral Definition and 125 Threads

\int_{-\infty}^{\infty}\frac{\ln{(a+ix)}}{x^2+1}dx I tried with integration by parts but go nowhere. I think it may require a branch cut and integrating along a contour. How would you approach this?

write f(z) in terms of partial fractions and integrate it counterclockwise over the unit circle where f(z)=\frac{2z+3\iota}{z^2+\frac{1}{4}} \\ 2\int\frac{z}{z^2 + \frac{1}{4}} dz \ + \ 3\iota \int\frac{1}{z^2 + \frac{1}{4}}dz \\ \ \mbox{. Also } \ z(t) = \exp{\iota t} (0 \leq \ t \...

Evaluate \int_C f(z)dz by theorem 1 and check your result by theorem 2 where f(z) = z^4 and C is the semicircle |z|=2 from -2i to 2i in the right half-plane. Theorem 1 : \int_{z_0} ^{z_1} f(z)dz = F(z_1) - F(z_0) \ \ \frac{dF}{dz}=f(z) \\ \int_{-2\iota} ^{2\iota}z^4dz = \frac{z^5}{5} =...

Homework Statement Let \gamma_{r} be the circle centered at 2i with a radius r. Compute: \oint \frac{f(z)}{z^{2}+1}dz Homework Equations 2 \pi i f(w)=\oint \frac{f(z)}{z-w}dz Cauchy's integral formula... maybe? The Attempt at a Solution I can see how to find solutions...

Homework Statement Evaluate: \int _{c} \dfrac{1- Log z}{z^{2}} dz where C is the curve: C : z(t) = 2 + e^{it} ; - \pi / 2 \leq t \leq \pi / 2 Homework Equations I know the independance of path in a domain where f(z) is analytical, but I tried the standard parametrization...

Homework Statement Homework Equations I hope there's someone who can help me with the following: I have to calculate the integral over C (the unit cicle) of (z+(1/z))^n dz, where z is a complex number. The Attempt at a Solution I tried to use the subtitution z=e^(i*theta), so...

Homework Statement \oint _{|z+i|=1} \frac{e^z}{1+z^2} dz =? The Attempt at a Solution I substituted z+i=z' and [itex]z'=e^{i\theta}[/tex] to arrive at e^{-i} \int _0 ^{2 \pi} \frac{e^{e^{i \theta}}}{-ie^{i \theta}-2} d \theta I have no clue how to solve such an integral, any...

Here is the exercise: Use the indefinite integral to compute \int_{C} \sqrt{z}dz where C is a path from z = i to z = -1 and lying in the third quadrant. Note: \sqrt{z} = e^{(1/2)lnz} where the principal branch of lnz is defined on C \setminus [0,\infty]. ------- I am just a little unsure of...

It's quite a "strange" thing..why people have so many difficulties with dealing with integrals of the form: \int _{c-i\infty}^{c+i\infty}dsf(s)e^{st}=I(t) ? You can always make the change of variable s=c+ix so you get: \int _{-\infty}^{+\infty}dxf(c+ix)e^{ixt}=I(t)e^{-ict} (2)...

Hello, I have a question about a complex integral. The question is about the index of a curve. This curve is defined as: j = j1 * j2 with j1: r*exp(i*t) with t: [0,pi] and j2: [-r,r] This is quite simple: a half cirle followed by a line from (-r,0) through (0,0) to (r,0). To...

Hello, I have a question about a complex integral. The question is about the index of a curve. This curve is defined as: j = j1 * j2 with j1: r*exp(i*t) with t: [0,pi] and j2: [-r,r] This is quite simple: a half cirle followed by a line from (-r,0) through (0,0) to (r,0). To...

Hey, I'm about to do an exam tomorrow and I seem to be a little stuck on how to answer this problem from a past paper. We've mainly been integrating by using the residue theorem, I forgot how to do something like this: \int_{C} \left( z^2 - 1 \right)^\frac{1}{2} dz C = \{ z : |z - 10| = 25\}...

We have to integrate [(lnx)^2)/(1+(x^2)) from zero to infinity. I have set up the correct complex integral with a branch cut along the negative y-axis, but I end up with an integral of [(lnx)^2](1+x^2) from minus infinity to zero. I'm not sure how to deal with this.

I'm working through an example regarding the spectral content of a truncated wave, and came across this in the textbook. I have no idea what they mean by the cosine being even and the sine odd. If anyone can explain this step to me that would be great.

I'm studying for a math exam for Complex Variable Analysis. We had a bunch of problems, and we keep having to let an integral just equal zero, and we're not sure why, but it always works. What we do is this... we're integrated over a closed contour. and we're supposed to chose a point z_0...

let be the integral..where \zeta(s) is the Riemann zeta function. \int_{c-i\infty}^{c+i\infty}ds\zeta(s)(x^{s}/s) then what would be the result?..there would be two singularities at the points s=0 and s=1 the problem is if there would be any other singularitiy on the integral

I attempted to prove the following equality, but to no avail. Anyone is willing to lend a hand? \int_0^{\infty} s^{2t-2v} e^{i w s} ds + \int_0^{\infty} s^{2t-2v} e^{-i w s} ds = \left[ \left( \frac{1}{-iw}\right)^{2t-2v+1} + \left( \frac{1}{iw}\right)^{2t-2v+1} \right] \Gamma(2t-2v + 1)...

I'm trying to perform the following integral \pi \int\limits_0^\pi {e^{2x} } \left( {\frac{1}{2} - \frac{1}{2}\cos 2x} \right)dx I split the integral and temporarely ignore the Pi so that I get \frac{1}{2}\int {e^{2x} dx} - \frac{1}{2}\int {e^{2x} \cdot \cos } \left( {2x} \right)dx...

Hi, I'm doing the following as an exercise to try and get my head around complex numbers. Specifically, I need to understand what it means to take the natural log of a complex number and what it involves. Say I wanted to integrate 1/ (1 +x^2) dx I know this is arcTan(x). I can also...

In my book QM book, Gasiorowicz says that \int_{-\infty}^{\infty} e^{i(p-p')x/\hbar}dx = 2\pi \hbar \delta(p-p') Where does that come from? I mean, set i(p-p')/h = K. Then the solution is \frac{\hbar e^{i(p-p')x/\hbar}}{i(p-p')} and evaluate at infinity, it doesn't exists as the...

Now I have to evaluate \int_{-\infty}^{\infty} e^{-Bx^2} e^{-iAx} dx Splitting it in two using Euler's identity show that the imaginary part is 0 (cuz integrand is odd). Remains the real part 2 \int_0^{\infty} cos(-Ax) e^{-Bx^2} dx for which integration by parts leads nowhere.

\int \frac {1}{x\sqrt{4x+1}}dx Here's what I have done so far on this problem I let u= \sqrt{4x+1} , so then u^2=4x+1 , du= \frac {2dx}{u} and x= \frac {u^2-1}{4} Substituting, I get \int \frac {1}{(\frac{u^2-1}{4})u}du Then moving stuff around, I get 4 \int \frac...

complex integral... let be the integral \int_{-i\infty}^{i\infty}\frac{1}{exp(s)-1}ds then their poles are 2n\pi my question is How would we calculate this integral? i think that the contribution from the poles is -{\pi}Res(z_0) the main problem i find is when i make the change of...

Hi, Does this complex contour integral equal 0? [Int] (z^2)/(sin z) dz along the closed contour e^2(pi)(i)(t) 0<t<1 It should equal zero cause its analytic in the domain around te curve and the zero in the numerator is of higher order than the zero in the denominator at the point z=0...

Here is the problem: {\mathop{\rm Im}\nolimits} \int {e^{x(2 + 3i)} } dx One sec, I'm having another go at it. = {\mathop{\rm Im}\nolimits} \int {e^2 } e^{3ix} dx = {\mathop{\rm Im}\nolimits} \int {e^2 } [\cos (3x) + i\sin (3x)]dx \begin{array}{l} = \frac{{ - e^2 \cos...