residue

Hello Let $p$ be a rational prime such that $p\equiv 3\mod 8$ and let $\mathfrak p$ the unique prime ideal of $\mathbb{Q}(i)$ laying over $p$. Applying the norm residue properties i get : 1) $\left(\frac{2,p}{\mathfrak p}\right)=\left(\frac{2}{\mathfrak p}\right)^{v_{\mathfrak p}(p)}=...

We have a very large modulus integer n also we have very large number y we know that y is a quadratic residue modulus n.Also we know all 4 square roots of y. What is the best way of prime factorization of n ?

There is one problem I can't solve for the particular section under prime power modulus in my textbook, and it's this: If the reduced residue classes $a$ and $b \,($mod $p$) both have order $3^j,$ how can I show that the two residue classes $ab$ and $ab^2$ one of them has order $3^j$ and the...

Hi all, I am really finding it hard to apply the simple formula provided by the Residue Theorem to the following integral \int_{-\infty}^{0}\frac{\alpha h(\alpha) Ai'(z_0) e^{i \alpha x}}{(i \alpha)^{1/3} \alpha \int_{z_0}^{\infty} Ai(z)dz + Ai'(z_0)}d\alpha where h is a function of...

Calculate $$\oint_c\frac{1}{z^2+9}dz$$ We can rewrite it as $$\oint_c\frac{1}{(z+3i)(z-3i)}dz$$ and we see that we have two poles, at $z=3i$ and at $z=-3i$. Residue theorem says: $$\int_\gamma f(z) dz=2\pi i\sum Res_{z=\alpha_{k}}(f)$$ So in order to calculate the the integral, we...

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^n≡a(m) , if m has a primitive root g, if (a,m) are relatively prime, then a solution exists iff a^{ϕ(m)/d}≡1(m) with d=(n,ϕ(m)) ϕ(m) is euler's...

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

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??

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...

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...

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 \...

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...

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

Find the last digit number of 2013^2013^2013^...^2013. Where the powers are 2013 times.

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?

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...

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

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?

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...

(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...