Recent content by Kate2010

Homework Statement Prove that \int^{∞}_{-∞} exp(-(z-ia)2)dz = √∏ for all real a. Homework Equations The Attempt at a Solution If I use the substitution x = z-ia then dz = dx and if I use the limits x = -∞ to x = ∞ I get the correct answer. However, I do not know how to justify leaving the...

Thanks :)

Homework Statement Find the Riemann function for uxy + xyux = 0, in x + y > 0 u = x, uy = 0, on x+y = 0 Homework Equations The Attempt at a Solution I think the Riemann function, R(x,y;s,n), must satisfy: 0 = Rxy - (xyR)x Rx = 0 on y =n Ry = xyR on x = s R = 1 at (x,y) = (s,n) But I...

Homework Statement Let L = Q(t) be the field of rational functions with one variable over Q. Consider the field automorphisms of L defined by a : t -> 1 - t and b : t -> 1/t . Find the relations. I will then be using this to find the size and abstract structure of the subgroup G of Aut(L)...

Thanks guys - I was trying to make things more complicated than they were.

Homework Statement If a>1 is a product of distinct primes, show that xn-a is irreducible over Q for all n ≥ 2. Homework Equations The Attempt at a Solution I am not really sure how to start this problem. Can anyone point me in the right direction? I know tests for...

But then doesn't that give g(f(x)) = x1 instead of f(g(x)) = x1?

Ok, thanks! How about F(x1,...,xn) = (f(x1,...,xn), x2,..., xn)? Then DF(0) is a matrix that is upper triangular, with 1s on the diagonal except for the first diagonal which we know is non-zero, then the det is non-zero, so it is invertible. Also we know that F(0,...,0) = (0,...,0). So by the...

Homework Statement Let f \in C1(Rn) be a function such that f(0) = 0 and \delta1f(0) is nonzero. (\delta1 means partial derivative with respect to x1) Show that there exist neighbourhoods U and V of x=0\in Rn and a diffeomorphism g:U->V such that f(g(x)) = x1 for all x = (x1,...,xn) \in U...

Ah yes :) thanks!

Thanks :) I get it now.

Homework Statement Use contour integrals and justify your steps, find \int cos2nx dx where the integral is from 0 to 2pi. Homework Equations The Attempt at a Solution My first thought was that this integral would be zero, using the residue theorem with residue 0, as I'm...

Homework Statement I'm revising at the moment and a bit stumped on question 4 http://www.maths.ox.ac.uk/system/files/attachments/AC104.pdf Homework Equations The Attempt at a Solution I think for the first part of the question, the regular singular points are 0 and -2...

Sorry if I am being dumb, but surely when I use y(x) = A(1-exp(-x)) and try to satisfy the boundary condition y'(1) = 0, I get y'(x) = Aexp(-x) so y'(1) = Aexp(-1) so A = 0?

Homework Statement Find a green's function G(x,t) for the BVP y'' + y' = f(x), y(0) = 0, y'(1) = 0. Homework Equations The Attempt at a Solution I solved the homogeneous equation, looking for 2 linearly independent solutions, and found A (constant) and exp(-x). I am struggling...