Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 3 / a

For the real decaying Bessel form used here, take

k > 0

. The original PDF has

K_{m} (r)

as the last factor in its auxiliary differential equation, although both differentiated factors have argument

k r

; that last argument must also be

k r

Put

B (r) = K_{m} (k r)

. The corrected identity is

B^{''} = - B^{'} / r + (k^{2} + m^{2} / r^{2}) B

. Since the field is radial, the radial Laplacian gives

(Δ - k^{2}) (r^{- p} B) = r^{- p} [B^{''} + \frac{2 - 2 p}{r} B^{'} + (\frac{p ( p - 1 )}{r ^{2}} - k^{2}) B] = r^{- p} [\frac{1 - 2 p}{r} B^{'} + \frac{p ( p - 1 ) + m ^{2}}{r ^{2}} B] .

Away from the point source this must vanish. For

k > 0

, a constant ratio of

r B^{'}

B

would make

B

a pure power, incompatible with the

k^{2} r^{2}

term of its equation. Therefore both displayed coefficients must vanish:

p = \frac{1}{2}, m^{2} = \frac{1}{4}, so m = \pm \frac{1}{2} .

K_{- 1/2} (z) = K_{1/2} (z) = π / (2 z) e^{- z}

The source describes

k

merely as real, but the following displayed real square-root formula implicitly needs

k > 0

. For the operator with arbitrary real

k

, its decaying fundamental solution instead uses

κ = ∣ k ∣

;

k = 0

is obtained by a limit. The unmodified real

A K_{m} (k r)

ansatz is not a real Bessel formula for all negative

k

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