13E (Further Complex Methods)

p T = κ T_{xx} .

Decay as

x \to \infty

selects

T (x, p) = f (p) e^{- x p / κ} .

The boundary value gives

f (p) = a (\frac{1}{p} - \frac{1}{p + b}) = \frac{ab}{p ( p + b )},

T (x, p) = \frac{ab}{p ( p + b )} e^{- x p / κ} .

b

Use the principal square root and deform the Bromwich contour around the negative real-axis branch cut. The pole at

p = 0

contributes

a

. The two boundary values at

p = - b

combine to

- a e^{- b t} cos (x b / κ) .

On the cut put

p = - v^{2}

; the jump of

e^{- x p / κ}

- 2 i sin (xv / κ)

and

d p = - 2 v, d v

. Taking the principal value at

v = b

gives

T (x, t) = a [1 - e^{- b t} cos (b / κ x)] + \frac{2 ab}{π} P \int_{0}^{\infty} \frac{e ^{- v^{2} t} sin ( xv / κ )}{v ( v ^{2} - b )} d v .

The expression has the prescribed boundary value and decays into the bar.

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