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68 Lectures on Scientific Subjects

A necessary and sufficient condition that, to a general har-
monic continuous density function of the mth order,

F(r, θ) = Fm(f) cos mθ+Gm(f) sin mθ,
there corresponds a continuous distribution function [unique
and harmonic of the same order m by what precedes},
f(s, φ) ≈fn(s)
cos mφ+gm(s') sin

is that the integral equation for hm(s^),

(3)             Hm(f) = hm(r sin u)e"mudu

where

Hm(f) = Fm(r) +iGm(r~),
admits of a continuous solution, whose real and imaginary
coefficients will then yield fm(s^) and gm(s} respectively.

The proof of this italicized statement follows immediately
from (1) and what has been proved in the preceding section.

7∙ EXPLICIT SOLUTION IN THE HARMONIC CASE Wl = I

In the harmonic case m = 1 we have to consider the integral
equation

(3ι)                H1(F) h1(r sin u)e~"idu,
which expresses the relation between the known coefficients
F1(F), G1(F) of the density function F and the like coefficients
/i(ʃ), gɪ(ʃ) of the distribution function
f(s, φ^) which it is
desired to find. This equation may be written

7Λ(r) =f* h(r sin w)(cos и — i sin u)du.

But the first component of the integral on the right is clearly

ɪʌ/'ɔ (r sin u)


25Γ

=0,


while the second term evidently has the value



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