Problem Sets
Problem Set 1 ( due Sep 12, Mon)
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Chapter 1, Prob 1a, 1c;
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Chapter 1, Find the general solution of
(D-1)(D-2)(D-3)y=$\exp(-x^{2})+x.$
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Chapter 1, Prob 2a, 2c.
Problem Set 2 ( due Sep 19, Mon)
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Consider the linear differential equation
$y^{\prime\prime\prime}+a(x)y^{^{\prime\prime}}+b(x)y^{\prime}+c(x)y(x)=f(x).$
Let the complementary solutions be $Y_{1}(x),$ $Y_{2}(x)$ and $Y_{3.}(x).$ The Wronskian $W(X)$ for this equation is the determinant
$\begin{array} [ Y_{1}(x) & Y_{1}^{\prime}(x) & Y_{1}"(x)\\ Y_{2}(x) & Y_{2}^{\prime}(x) & Y_{2}"(x)\\ Y_{3}(x) & Y_{3}^{\prime}(x) & Y_{3}"(x) \end{array}$.
Show that
$W(x)=c\exp[-\int_{0}^{x}a(x^{\prime})dx^{\prime}],$
where $c$ is a constant.
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Prob 1 in Chapter 2 (p.111).
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Prob 2 in Chapter 2 (p.111).
Problem Set 3 (due Sep 28, Wed)
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Prob 1. Evaluate the following integrals:
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$\int_{-\infty}^{\infty}\dfrac{dx}{(x^{2}+9)(x-3i)(x-5i)}.$
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$\int_{0}^{2\pi}\dfrac{d\theta}{(2+\sin\theta)^{2}}.$
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Prob 2. Evaluate the integrals in Problem 4c and 4d on p. 112.
Problem Set 4 (due Oct 12, Wed)
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Prob 1. Evaluate the integral
$I=\int_{-\infty}^{\infty}\dfrac{1+\cos\pi x}{1-x^{2}}dx$
in two ways:
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Deforming the contour of integration away from the points $z=\pm1.$
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Expressing the integral as the real part of
$J=P\int_{-\infty}^{\infty}\dfrac{1+e^{i\pi x}}{1-x^{2}}dx,$
where $P\ $ denotes the principal value of the following integral.
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Problem Set 5 (due Oct 21, Fri)
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The Klein-Gordon equation with a source term $\rho(\overrightarrow {x},t)$ is
$(\dfrac{\partial^{2}}{\partial t^{2}}-\bigtriangledown^{2}+m^{2})\phi(\overset{\rightarrow}{x},t)=\rho(\overrightarrow{x},t)$ , $\left\vert \overset{\rightarrow}{x}\right\vert <\infty,$ $t>0,$ $m$ a constant.
Find the solution of this equation which satisfies the initial conditions
$\phi(x,0)=f(x),$ $\phi_{t}(x,0)=g(x),$
and vanishes at the spatial infinities. Identify from this solution the Green functions for the equation.
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Find the Green's function of the Klein-Gordon equation $G(\overset{\rightarrow}{x}-\overset{\rightarrow}{x^{\prime}},t-t^{\prime})$ by solving
$(\dfrac{\partial^{2}}{\partial t^{2}}-\bigtriangledown^{2}+m^{2})G(\overset{\rightarrow}{x}-\overset{\rightarrow}{x^{\prime}},t-t^{\prime})=\rho(\overset{\rightarrow}{x}-\overset{\rightarrow}{x^{\prime}},t-t^{\prime})$
with appropriate initial conditions. Show directly how to solve the problem in Prob 1 with the Green finction.
Problem Set 6 (due Nov 7, Mon)
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Let $y$ satisfy the equation
$y"+x^{2}y=0$
and the initial conditions $y(x_{0})=1,$ $y^{\prime}(x_{0})=0$ where $x_{0}>0.$
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Find the WKB approximation of $y(x)$ for $x$ > $x_{0}$. For what values of $x$ do you expect it be a good approximation?
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Use the computor to obtain the numerical values of $y(x)$ as a function of $x$ for $x_{0}=1,5,10.$ Compute also the numerical values of the WKB approximation of $y(x)$. Compare the computor result with the approximate result.
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Problem 4, Chapter 7.
Problem Set 7 (due Nov 21, Mon)
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Plot the integral
$I(\lambda)\equiv\int_{-\infty}^{\infty}$exp($-\lambda\sinh^{2}x$) $dx$
for $\lambda>0.$
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Plot$\sqrt{\pi/\lambda},$the aproximatate value of this integral obtained via the Laplace method and compare with the result you get in (a).
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Find the leading term for the following integrals for $\lambda>>1$:
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$\int_{0}^{\pi/2}e^{-\lambda\cos t}dt,$
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$\int_{-1}^{1}e^{\lambda t^{2}}dt,$
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$\int_{-\infty}^{\infty}e^{\lambda(x^{2}-x^{4})}dx.$
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The Laplace transform of the function $f(x),$ $0$ < $x$ < $\infty,$ is defined as
$L(s)=\int_{0}^{\infty}dx$ $e^{-sx}f(x).$
Find the asymptotic form of $L(s)$ when $s>>1.$ Verify your result with the examples of $f(x)=\sin x$ and $f(x)=\cos x.$
Problem Set 8 (due Nov 30, Wed)
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Find the asymptotic form for
$I(k)=\int_{-\infty}^{\infty}e^{-ikx}e^{-x^{4}}dx,$ $k>>1,$
(The integral above is the Fourier transform of $e^{-x^{4}}).$