# Lecture Notes

### Site Tools

math121b:midterm2-solution

# Midterm 2, Solution

## 1. Gamma functions and Beta function (15 points)

You may use Gamma or Beta function to express the final answer. Please show intermediate steps, otherwise there are no points.

1. Compute the integral (5 points) $$\int_0^1 \frac{x^4}{\sqrt{1-x^3}} dx$$

Replace variable $u = x^3$, then $$I = \int_0^1 \frac{u^{4/3}}{(1-u)^{1/2}} du^{1/3} = (1/3) \int_0^1 (1-u)^{-1/2} u^{2/3}du = (1/3) \int_0^1 (1-u)^{-1/2} u^{2/3}du = (1/3)B(5/3, 1/2).$$

2. Compute the integral (5 points) $$\int_0^1 x^2 (-\ln x)^2 dx$$ Replace variable $u = -\ln x$, then $$\int_0^\infty e^{-2u} u^2 e^{-u} du = (1/3)^3 \Gamma(3).$$

3. Compute the special value of Gamma functions (the result should not be expressed using Gamma function).

• $\Gamma(-1/2) = \Gamma(1/2)/(1/2) = 2 \sqrt{\pi}$ (2 points)
• $|\Gamma(1/2 + i)| = ?$ (3 points) [Hint: $\Gamma(\bar z) = \overline{\Gamma(z)}$, and $\Gamma(p)\Gamma(1-p) = \pi / \sin(p \pi)$ ]

Take $p=1/2+i$, then $$|\Gamma(1/2 + i )|^2 = \Gamma(1/2+i) \Gamma(1/2-i) = \pi / \sin(\pi (1/2 + i)) = \pi / \cos(i \pi) = \pi / \cosh(pi)$$

## 2. Legendre Function (25 points)

1. Compute $P_3(x)$ using Rodrigue formula. (5 points)

By Rodrigue formula $$P_3(x) = \frac{1}{2^3 3!} \d_x^3 (x^2-1)^3 = \frac{1}{2^3 3!} \d_x^3 (x^2-1)^3 = \frac{1}{2^3 3!} \d_x^3 (x^6- 3 x^4 + 3 x^2 - 1) = \frac{1}{48}(120 x^3 - 72 x) = (5 x^3 - 3 x)/2.$$

2. Prove the recursion relation 5.8( c) (10 points) $$P_l'(x) - xP_{l-1}'(x) = l P_{l-1}(x)$$ using the generating function $$\Phi(x,h) = \sum_{n=0}^\infty h^n P_n(x) = \frac{1}{\sqrt{1 - 2 x h + h^2} }$$

If we multiply both sides of the equation by $h^l$ and sum over $l$, we get $$LHS = \d_x \Phi(x,h) - x h \d_x \Phi(x,h)$$ $$RHS = h \d_h (h \Phi(x,h))$$ We plug in $\Phi(x,h)$ to test whether this is true. $$LHS = (1-xh) \frac{h}{(1 - 2 x h + h^2)^{3/2}}$$ and $$RHS = h \Phi + h^2 (-1/2) (-2x+2h) (1 - 2 x h + h^2)^{-3/2} = [h(1-2xh + h^2)+h^2(x-h)](1 - 2 x h + h^2)^{-3/2}$$ $$= h(1-hx)(1 - 2 x h + h^2)^{-3/2}$$ OK, the same, done.

3. Compute $\int_{-1}^1 x^n P_n(x) dx$ in the following steps: ( 10 points)

• Find the constant $c$, such that $P_n(x) = c x^n + \z{ lower order terms}$. (try Rodrigue formula to get the leading term)
• Show that $\int_{-1}^1 P_n(x)^2 dx = \int_{-1}^1 c x^n P_n(x) dx$
• Look up $\int_{-1}^1 P_n(x) P_n(x) dx$.

The leading term of $P_n(x)$ can be obtained as $$\frac{1}{2^n n!} (2n)(2n-1)\cdots (n+1) = 2^{-n} {2n \choose n}$$

Because lower order term is orthogonal to $P_n(x)$.

$\int_{-1}^1 P_n(x) P_n(x) dx = \frac{2}{2n+1}$

So finally, we get $$\frac{2^{n+1}(n!)^2}{(2n+1) (2n)!}$$

## 3. Bessel Function (30 points)

1. Problem 12.1. Show by ratio test that the series for $J_p(x)$ converges for all $x$. (10 points)

2. Problem 15.6 (7 points)

3. Problem 19.1 (7 points)

4. Problem 20.3, 6, 7 (6 points)

## 4. Solving PDE with separation of variables (30 points)

1. Solve the steady state heat equation on 2D square $[0,1]^2$. (10 point) $$\Delta u(x,y) = 0, \quad 0 \leq x, y \leq 1$$ with boundary condition that $$u(0, y)= 0, u(1,y)=1, u(x, 0)=0, u(x,1)=1.$$

Solution $$u(x,y) = \sum_{n \z{ odd} } \frac{4}{n \pi \sinh(n \pi)} \left( \sin(n \pi x) \sinh(n \pi y) + \sin(n \pi y) \sinh(n \pi x) \right)$$

2. Solve the steady state heat equation on 3D unit ball. (10 point) $$\Delta u(r, \theta, \phi) = 0$$ with boundary condition at $r=1$ that $$u(1, \theta, \phi) = \cos(\theta) \sin(\theta) \sin(\phi)$$

Hint: use $P_2^1(\cos \theta) = -3 \cos(\theta) \sin(\theta)$

Solution: $$u = (-1/3) r^2 P_2^1(\cos(\theta)) \sin(\phi)$$

3. Solve the heat flow equation on a circle. (10 point) $$\d_t u(t, \theta) = \d_\theta^2 u(t, \theta).$$ such that the initial condition is $$u(0, \theta) = \cos^2(\theta).$$

$$u(\theta, t) = e^{-4t} (1/2) \cos(2\theta) + 1/2$$ 