Recent content by ab959

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    Complex integral via complex contour

    Hi, I am using the Fourier transform to price a European put option. I have obtained the following integral: q(x,\tau)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{-[\omega^2-i\omega(\gamma-3)+2(\gamma-1)]\tau}}{(1-i\omega)(2-iw)}d\omega which I need to solve. I have solved this through...
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    Quartic with complex coefficients

    OK great thanks I realised this soon after I posted. What I am more interested in knowing is if you classify the roots in terms of the parameters. I.e. knowing when there will be 4 real roots or complex roots etc. My intuition tells me however that if Im(p) not equal to zero then all...
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    Quartic with complex coefficients

    I am trying to solve a fourth order polynomial which is in the following form x^4+Ax^3+(B_1+B_2p)x^2-(C+Ap)x+D+Ep=0 Where A, B_1, B_2, C, D, E, are real parameters and p is a complex parameter. I have investigated many ways of solving this equation however there does not seem to be...
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    Second-order Coupled O.D.E.s with constant coefficients

    1. Homework Statement I am trying to solve a system of two coupled ODEs. I am interested in an analytic solution if that is possible. I know it will be messy. \frac{\partial^2 U_1}{\partial x^2}+a_1\frac{\partial U_1}{\partial x^2}+b_1 U_1 = c_1 U_2 \frac{\partial^2 U_2}{\partial...
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    Tricky semi-infinite integral

    I ended up just using L'Hopital's rule and it came out quite easily. Thanks Tom you were a lot of help! Very much appreciated.
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    Tricky semi-infinite integral

    Hi Tom. As my b is a real number, for the case b=0 this doesn't work. I want to find out the limit as b-> 0 of b * integral. I know that it is equal to zero but I am having proving this. All I really have to do is show that the integral with b=0 is finite. Any suggestions?
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    Tricky semi-infinite integral

    Cool. Thanks for the help! It was a very nice step. Here is the solution for anyone interested: \int^\infty_0 \frac{e^{-tp^2}}{{p^2+b^2}} cos(pu) dp = \frac{\pi e^{tb^2} cos(ibu)}{2|b|}}
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    Tricky semi-infinite integral

    Yes I am familiar with it however not in this form. Are you suggesting forming a curve on the complex plane which encloses the points p=+/-ib. The integral over this curve would equal 2*pi*i* Sum of residues?
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    Tricky semi-infinite integral

    1. Homework Statement I am trying to integrate \int^\infty_0 \frac{e^{-tp^2}}{{p^2+b^2}} cos(pu) dp. 2. Homework Equations I know that \int^\infty_0 \frac{e^{-tp^2}}{{p^2+b^2}} dp= \frac{\pi}{2b}e^{tb^2}erfc(\sqrt{a}x) 3. The Attempt at a Solution I rewrote the problem in terms of the...
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    Help! Two fused semi-infinite rods - hard integral

    1. Homework Statement The problem is to find an function that describes the tempurate variation with time and position of an infinitely long heated rod that is made by fusing together two semi infinite rods at x=0. They have perfect thermal contact but are made of two different materials...
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    Heated rod problem

    1. Homework Statement I was asked to write the equation that describes the following heated rod: Two rods of different materials that have been fused together on the x-axis at x=0 to form one rod of infinite length. I was thinking of just using the heat equation integrated from...