2A (Differential Equations)

Start from

I_{0} (λ) = 1/ λ

I_{n}^{'} (λ) = - I_{n + 1} (λ), I_{n} (λ) = (- 1)^{n} \frac{d ^{n}}{d λ ^{n}} \frac{1}{λ} .

This interchange is justified locally around any

λ > 0

by the dominated convergence theorem: the differentiated integrands are bounded by an integrable polynomial times

e^{- λ x /2}

. Successive derivatives of

λ^{- 1}

give

I_{n} (λ) = \frac{n !}{λ ^{n + 1}} .

This is also the integer case of the Gamma function integral after scaling the integration variable.

b

u = (x + y) /2

and

v = (x - y) /2

\partial_{x} = \frac{1}{2} (\partial_{u} + \partial_{v}), \partial_{y} = \frac{1}{2} (\partial_{u} - \partial_{v}) .

The mixed partial derivatives cancel, assuming the twice continuously differentiable solution appropriate here. Consequently

f_{x y} = (g_{uu} - g_{vv}) /4

The transformed equation is

g_{uu} - g_{vv} = 4.

(2)

The factors of

1/2

belong to the inverse Jacobian matrix; using the forward transformation instead would give an incorrect factor.

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