Midterm 1 Solutions, Physics 114B

10/17/01

in

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 $z=0$. Expanding numerator about $z=0$: $1-\cos z \propto z^2$ so $(1-\cos z)^2 \propto z^4$ therefore $(1-\cos z)^2/z^3 \propto z$. Therefore it's regular.

(b)(10 points) ${1\over 1-\exp(z^3)}$, at $z=0$. Expanding denominator about $z=0$: $1-\exp(z^3) \approx -z^3$, so pole is third order.

(c)(10 points) $\sin({1\over \cos z})$, at $z=\pi/2$. Expanding $\cos$ about $\pi/2$, $\cos z \approx -(z-\pi/2)$. $\sin({-1\over z-\pi/2})$ has an essential singularity.

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

2. Calculate

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

by doing contour integral of

$$\int_0^{\infty} {1\over 1 + z^4} dz$$

by completing it with the usual semi-circle in upper (or lower) half of the complex plane. We know that the integral along the semi-circle vanishes as the radius of the circle goes to infinity, as discussed in the text.

There are 4 poles at $z = e^{i\pi/4 + 2\pi n/4}$ with $n = 0,1,2,3$. In the upper half plane there are two poles at $z_{\pm} = (\pm 1 + i)/\sqrt{2}$.

Applying the residue theorem:

$${\int_0^{\infty}}_C {1\over 1 + z^4} dz = 2\pi i (\sum Res {1\over 1 + z^4})$$

Residues most easily calculated using the derivative trick (Boas Chapter 14, eqn. (6.2)) where we differentiate denominator. That gives the residue of the function ${1/(4z^3)}) = z/(4z^4)$. But $z^4 = -1$ at all poles so this is then $-z/4$. Therefore integral $I$ equals

$$2\pi i (-z_- -z_+)/4 = 2\pi i {1\over \sqrt{2}} ((1+i) + (-1+i))/4 = \pi/\sqrt{2}.$$

3. The same as Boas Chapter 13 section 3, except with a different initial condition and diffusion coefficient $D = \alpha^2$. So (a) through (d) are identical to the text. However for (e) instead of a linear temperature ramp, it is at uniform temperature $T_0$. So we need to do Fourier series for that case instead. Boas does this in Chapter 13 Eqn (2.11).

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