Recent content by jjangub

Homework Statement How do I change sin^2(z) to x+iy form? (z=x+iy) I have to put this x and y to arctan(y/x) Homework Equations The Attempt at a Solution I tried to use sin^2(z) = 1/2 -1/2(cos(2z)) or sin(z) = ((e^(iz) - e^(-iz))/2i)^2 but both ways I cannot take out i. Or isn't the...

well, if i do it again, i got z = e^i(0) = 1 and z = e^i(2pi) = cos(2pi) + isin(2pi) = 1 is this right?

so...path is 0 to infinity? so I have to multiply by 1/2 to 3pi/2? well...this cauchy and residue theorem parts are so confusing for me... in class, prof gave us the easy example, so I understood, but the question for assignment and tests are hard... I have test next week...how should I...

well...I tried... but how do I get 3sqrt(3)? I can't even get any roots... some hints please...

maybe this will be the last question...I hope so if I convert polar integral (with cos and sin) to complex integral, then it changes from integral(zero to 2pi) to integral(negative infinity to positive infinity)? or it works only if it is unit circle? so i end up with Res(z=-3i/4)f =...

yes, I made a mistake. Now, I got that equation. I know this quesetion sounds silly, but I am kind of lost on cauchy integral and residue theorem. so do we have different method for when we have 1 order of pole, 2 order of pole or higher, and for cos and sin? Is there any good site which is...

sorry, but can you explain with other formula? i didn't learn this formula, i instead learned an and bn to find it. but this function is different...

Homework Statement Evaluate \int_{0}^{2\pi} (cos^4\theta + sin^4\theta) d\theta by converting it to a complex integral over the unit circle and applying the Residue Theorem. Homework Equations The Attempt at a Solution First, I switch (cos^4\theta + sin^4\theta) to...

Homework Statement Show \int_{-\infty}^{\infty} 1/((x^2+x+1))^2 dx = 4\pi/(3sqrt(3)) Homework Equations The Attempt at a Solution First, I tried to find the singularities, I used (-b\pm\sqrt{b^2-4ac})/2a so I got two singularities, but only (-b+\sqrt{b^2-4ac})/2a works. Now, I got...

Homework Statement Find the Laurent series about 0 of 1/sinh up to (and including 0) the z5 term Homework Equations The Attempt at a Solution Since 1/sinh is equal to (1/z) * (1/(1+(z^2/3!)+(z^4/5!)+(z^6/7!)+...)) So if we work on the second term by dividing 1 by denominator and multiply...

I just learned residue today...is there any other way to do this? well, I can use residue, but I am not confident about it. I have to do this by tomorrow...

For the first term, it has a taylor series of 1-z+z^2-z^3+... and the coefficient of z is the desired residue, 2\pi*i*(-1) = -2\pi*i For the second term, taylor series is z+2*z^2+3*z^3+... and the coefficient of z is the desired residue, 2\pi*i*(1) = 2\pi*i so if I add these two then, I get...

so do I get zero as the answer? I tried residue theorem and I got zero.

I understand most of them excpet c). Could you explain again about c)? Thank you.

Homework Statement 1) Evaluate \intc ((5z-2)/(z(z-1)(z-3)))dz where c is the circle of radius 2 about the origin. 2) Evaluate \intc (2*(z^2)-z+1) / ((z-1)^2(z+1))dz where c proceeds around the boundary of the figure eight formed by two circles of radius 1 with centres 1 and -1 by starting at...