Midterm 1, Physics 114B

10/17/01

in Read the questions carefully. To receive credit, please provide at least a brief explanation of how you arrived at your answer. Do not write down multiple answers to the same problem.

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1. For each of the following functions, determine if the indicated point is regular, an essential singularity, or a pole, and if it is a pole state what order it is.

(a)(10 points) $(1-\cos z)^2/z^3$, at $z=0$.

(b)(10 points) ${1\over 1-\exp(z^3)}$, at $z=0$.

(c)(10 points) $\sin({1\over \cos z})$, at $z=\pi/2$.

(d)(10 points) $1/z$, at $z=1$.

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2. (40 points) Calculate

$$\int_0^{\infty} {1\over 1 + x^4} dx$$

using the residue theorem.

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3. Consider a rod of length $L$ and diffusion coefficient $D$. Initially it is at a uniform temperature of $T_0$ degrees. At time $t=0$ the ends are refrigerated so that their temperatures are held at $0$ degrees.

(a)(10 points Using separation of variables, $T(x,t) = {\cal X}(x){\cal T}(t)$, separate the diffusion equation

$$D{\partial^2 T\over \partial x^2} = {\partial T\over \partial t}$$

To obtain two ordinary differential equations, one for $\cal X$ and one for $\cal T$.

(b)(10 points) Find the general solution of both the two ordinary differential equations.

(c)(10 points) Use the boundary conditions at the two ends to obtain $\cal X$.

(d)(10 points) Write out the solution as the sum over all possible solutions obtained above. Your answer should involve an infinite number of coefficients $a_n$.

(e)(10 points) Use Fourier series to obtain the $a_n$'s by using the initial condition that $T(x,t=0) = T_0$.

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Useful information

$$\int_0^{L} \sin^2(n\pi x/L) dx = L/2$$

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