# residue

1. ### Complex analysis- residue theorem

Is it possible to solve the above using residue theorem? if yes then how to start?
2. ### Proof Residue 165 = 3 * 5 * 11?

Hi guys, I wanna ask if x = 1 (mod 3) 2x = 1 (mod 5) 3x = 1 (mod 11) How can I find all residue from 0-164, using the fact that 3*5*11 = 165? I tried to use the very basic way to start off x = 1+3k and substitutes and so on... but can anyone explain to me how can we use the fact tk solve the...
3. ### Use the residue to calculate the integral over the contour

Hi! I trying to calculate the integral, but without success. Can anyone help me with this ?
4. ### Residue and modulo

Hi guys, i am pretty confused with the topic... I need to find residue of 3^2015 modulo 11 using Fermat's Little Theorem, should I start off with 2^2 = 1 (mod 10) or what should I do... I have read the examples but I do not understand.. can someone please explain it to me Cheers
5. ### Find the least residue of

A) 62543^2173 mod 9 B) 12345^12345 mod 11 C) (15*16) * (7+22^32) mod 13
6. ### Find the least residue of the multiplicative inverse of a mod m

A) a=7, m=11 B)a=7, m= 23 C)a=5, m=31 D)a=117, m=24
7. ### Residue (Computing Integral along branch integral)

While computing this integral \int_{-1}^{1}\frac{dx}{\sqrt{1-x^2})}, I am having trouble at finding residue at -i. My work is like this a_{-1}(-i)=\frac{1}{\sqrt{2}.2e^{i3\pi/2}}, but there is mistake in here and I haven't been able to figure it out.
8. ### finding residue

how does one find residue for e^z/(1-cosz) = f(z) i'm looking integral of f(z) on circle of radius 2 located at origin i can't seem to figure out what method to use. it has been a long time since i've done comp analysis.
9. ### Residue theorem

Please refer to attached material. For the first question, I have tried looking at examples and have noted that the bounds have been provided in a manner: like |z|=1 (as given in part ii) I am not sure how to get transform the given |z-pi|=pi in such a format, although i suspect it would be...
10. ### Residue Modulo

1. Compute the least positive residue modulo 10,403 of 7651^891 2. Compute the least positive residue modulo 10,403 of 7651^20!
11. ### Definite Integration using Residue Theorem

Evaluate \int_{0}^{\infty} cos(x^2) dx. What complex integral and branch cut should I use??
12. ### Finding the residue of the following function

Hi all, I have the following function f(z) = \frac{1}{(z^2 + z + 1)^2} and I need to find the residue at the double pole z = (-1)^{\frac{2}{3}}. Any tips? I'm hoping to not have to use the Laurent expansion for this. I know that (z^2 + z + 1) = \frac{(1-z)}{(1-z^3)}. I'm aware of a limit formula...
13. ### Integral with residue, question about poles.

[Solved]Integral with residue, question about poles. Hello everyone, I am having a little problem with the following integral: \int \frac{sinz}{z^3(z^2+1)} Where D is the Domain D = {z in the complex field : |z|<2} Now, my understanding is I have to look where my function goes to zero. On...
14. ### cubic residue

p is an odd prime, and p does not divide u prove that if p is congruent to 2 mod 3, then every...

I'm having problems understanding part of a proof for determining when 2 is a quadratic residue. It's a corollary of Guass's lemma. Define a the P:={1,2,3,...,(p-1)/2} and N:={-1,-2,-3,...,-(p-1)/2} where p is an odd prime. Then 2P={2,4,6,...,(p-1)}. (*)Now if p=4q+1, then...

1) If ab=r mod p, where r is a quadratic residue of the odd prime p, prove that a & b are both quadratic residues of p or both non-residues of p. 2) If a and b are both quadratic residues of the odd prime p, or both non-residues of p, show that the congruence ax2=b mod p has a solution [hint...
17. ### Residue of a function

I need to find the residue of f(z) = 1/(sin(z)) at z=Pi ------------------------------------------------------------------------ I had a similar exercise for z=Pi/2. I found the expansion of f(z) to be 1/z + z/6 + ... then calculating the residue for a simple pole (?) i found that lim as...
18. ### Pls help me find the least non-negative residue of this problem..

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? I don't think Fermat's little theorem is applicable since 35 is not a prime. Can we use Euler's Theorem to solve this?
19. ### Find all odd primes p for which (11/p) = 1. Quadratic residue

Find all odd primes p for which \left(\frac{11}{p}\right) = 1. \left(\frac{11}{p}\right) is in Legendre Symbol. \left(\frac{11}{p}\right) = - \left( \frac{p}{11} \right) due to quadratic reciprocity. (11 \equiv 3 \pmod{4}) Since 11 is prime and p is prime, we have gcd(11, p) = 1. Apply Gauss'...
20. ### complete residue

is it right to say that if b is a primitive root mod p then {b^0,.....b^p-1} is a complete residue system mod p?