496 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
Remarks on Heine’s definition of P, 4 (p,). 
37. Heine has proposed* to define the function P„ (p.) for complex values of n and p, 
by means of the expression 
P„ fi) = " f + vV - 1 COS Xp) n dxji. 
IT J 0 
It will appear from what we have shown in Arts. 33-36, that this definition is not a 
valid one, as the function given by the definite integral for values of p. with a negative 
real part is not the analytical continuation of the function given by the same definite 
integral for values of p, with a positive real part; it follows that P„ (p.) can be defined 
by the above expression only for values of p. with a positive real part. 
The fact that the definite integral is of ambiguous meaning at the imaginary p, axis 
is clear if we attend to the phases of the integrand (p + — 1 cos v|/)", or h n ; /jl being 
purely imaginary there is a value of i fj between 0 and tt for which h vanishes, and in 
passing through this value of t p the phase of the integrand changes by a finite amount. 
The h integral in Art. 33 is taken along a path joining z, -which has the point h = 0 
on the left hand side, thus for purely imaginary values of p, the path may be placed 
T z 
—T- 
i 
z 
as in the figure, with a semi-circular portion to avoid the point h = 0 ; we thus see 
that in the above definite integral there must be a sudden diminution of phase mr in 
the integrand as cos \}j passes through the value ; if this be taken into account 
the definite integral will represent the function P„ (p) for purely imaginary values 
of p.; there is however nothing in the definite integral itself which decides apart from 
convention what the change of phase in the integrand shall be as it passes through its 
zero value. 
Next suppose p. to cross the imaginary axis, the h integral can then be taken from 
—- to z along a loop round the point h = 0, and then along a straight line to the 
* ‘ Kugelfunctionen.,’ Vol. 1, p. 37. 
