# residue

6. ### nth power residue

Hello, I've been trying to understand a proof about congruence but there's a passage that I cannot seem to get. given the congruence x^nâ‰¡a(m) , if m has a primitive root g, if (a,m) are relatively prime, then a solution exists iff a^{Ï•(m)/d}â‰¡1(m) with d=(n,Ï•(m)) Ï•(m) is euler's...
7. ### modulo 3^m residue detection in base 12

Hi I was trying to do this problem. But i am getting nowhere. Demonstrate that for natural number m and n, that the modulo 3^m residue of n can be determined from its last m digits when expressed in base 12. I am not sure what it is asking. Thanks in advance
8. ### Using Cauchy's residue theorem

Use Cauchy's residue theorem to evaluate the given integral over the indicated contour. \oint_C \frac{cos(z)}{(z-1)^2(z^2 +9)}dz ; C: |z-1|=1 So, w/ the indicated contour, I will only be concerned w/ the residue for 1, correct? Since \pm 3i is not w/in |z-1|=1??
9. ### Using Laurent series to find residue

Use the appropriate Laurent series to find the indicated residue. f(z) = e^{\frac{-2}{z^2}}, Res (f(z),0) The Laurent series I am using e^{\frac{-2}{z^2}}=1 - \frac{2}{z^2(1!)} + \frac{4}{z^4(2!)} - ...... From this, if this is correct, how do I find the residue? I know it is 0...
10. ### Residue calculation

Hi. I got a problem to find the residue at z=1 for the function f(z)=\frac{\cos{z}-1}{z^3(3z^2-27)}e^{\frac{i}{z-1}}. This seems pretty impossible. I can taylor expand the exponential, and I assume I have to expand everything else around z=1. This gets me nowhere.. Is there some smart way to do...
11. ### Find the residue

Im a little stuck on this one... Find the residue of f(z) given below at each of its poles Find the residue of f(z) given below at each of its poles f(z)=\frac{lnz}{(z^2+4)} Use the principal branch of the logarithm in you evaluation i.e Lnz = \, lnr \, + \, i \Theta \, , \, r \, > \, 0 \...
12. ### Evaluate the residue

Hi, Can anyone help me with this problem Evaluate the residue of f(z) = e^{-z}\frac{1}{(z-1)^4} at its isolated pole singularity In my text I found that an isolated singular point z_0 of a function f is a pole of order m if and only if f(z) can be written in the form...
13. ### Residue at the essential singularity

Hi, Im trying to work out this problem, can anyone give me a hand? Find the residue at the essential singularity of the integrand, and evaluate the integral below, where the contour is the positively oriented circle |z|=2 \oint_c zsin(\frac{1}{z^2}) \, dz
14. ### Residue and modulo

Find the last digit number of 2013^2013^2013^...^2013. Where the powers are 2013 times.
15. ### Integral Residue

Compute the limit along the given circular arc: \lim_r\to 0^+ \int_{T_r}\frac{2z^2+1}{z}dz, where T_r:z=re^{i\theta}, 0\le \theta \le \frac{\pi}{2}. The answer is: i(\frac{\pi}{2}-0) Res(0)=\frac{i\pi}{2}, but I do not know how they got that?
16. ### Residue Integral

Show that p.v.\int_{-\infty}^{\infty}\frac{e^{2x}}{\cosh(\pi x)}dx=\text{sec}1 by integrating \frac{e^{2z}}{\cosh(\pi z)} around rectangles with vertices at z=\pm p,p+i,-p+i. I am not sure how they got those vertices for the rectangle (can someone help clarify that?), but with those vertices I...
17. ### Modulo Residue problem

Hello All, I am stuck with this problem. How can I prove the series {kb( mod d)}, k = 1, 2, . . . , d contains d different residues Can anyone please help me in this regard. I will be grateful. Thank you
18. ### Pls help me find the least non-negative residue of this prob

How can you find the least non-negative residue of 2^20 modulo 35. If using a calculator, we can easily get 11, however, is there a concrete solution to show this? Can we use Euler's Theorem to solve this?
19. ### Residue theorem

I am recently reading about residue theorem and application, but looks I cannot understand those. Could you please help me solving some problems and understanding some concepts? What is the Pole, what are the zeros? Please post a step-by-step solution to the question: evaluate...
20. ### Mistake with a simple residue problem!

(First I will forget the 2{\pi}i that is required when calculating every residue as it is not important here) Calculate the residue at the simple pole z=-\frac{i}{2} for the function f(z)=\frac{1}{(2z+i)(z+2i)} If I use "The Residue at a Pole" lemma which states: Let \alpha be a pole of order...