The material in this question concerns substitution of a Schlaefli type integral into a differential equation. My answer to the question Showing Schlaefli integral satisfies Legendre equation should inform the interested reader.
Hobson$^1$, pg 183, tells us, that if we substitute
\begin{equation*} \nu= \int (t^2−1)^n(t−\mu)^{-n-m-1} ~dt \end{equation*}
into the left-hand side ( LHS ) of an equation labelled (3) from Hobson$^1$ pg 180, which is actually a transformed form of ‘Legendre’s Associated Equation’, see below
\begin{equation*} (1-\mu^2) \frac{ d^2 \nu }{ d\mu^2 } -2(m+1)\mu \frac{ d \nu }{ d\mu } +(n-m)(n+m+1)\nu = 0 \tag{3} \end{equation*}
we should find that (a quote from p183),
\begin{align*} \left\{ (1-\mu^2) \frac{ d^2 }{ d\mu^2 } -2(m+1)\mu \frac{ d }{ d\mu } +(n-m)(n+m+1) \right\} \int (t^2−1)^n(t−\mu)^{-n-m-1} ~dt \\ =-(n+m+1)\int \frac{d}{dt} \left\{ (t^2-1)^{n+1} (t-\mu)^{-n-m-2}\right\}~dt \end{align*}
Please note I have doubts about the minus sign at the beginning of the left-hand side above.
Hobson$^1$ then says that, quote
It thus appears that the differential equation (3) is satisfied by \begin{equation*} \nu= \int (t^2−1)^n(t−\mu)^{-n-m-1} ~dt \end{equation*} for unrestricted values of n and m, provided the integration is taken along a closed path of such a character that the integrand attains the same value, when the path has been completely described, as that with which it began. The integrand has, in general, the four singular points $t=1, t=-1, t=\mu, t=\infty $; a closed curve which contains in its interior one or more of these singular points will, in general, be such that the integrand attains a value after a complete description of the closed curve different from its initial value. The closed paths for which this is not the case , and which can therefore be employed to define an integral of the differential equation, may be characterized as being closed when drawn on the Riemann’s surface on which the function $(t^2−1)^n(t−\mu)^{-n-m-1}, ~of~t$, is represented as a single-valued function.
My question is: Should in the last sentence above,
\begin{equation*} (t^2−1)^n(t−\mu)^{-n-m-1} \end{equation*}
be replaced by \begin{equation*} ~~~~~~~(t^2−1)^{n+1}(t−\mu)^{-n-m-2}~~? \end{equation*}
Other
It also looks as if, in the large textual quote from Hobson$^1$, that
“such a character that the integrand attains the same value”
should be replaced by, something like,
“such a character that $(t^2−1)^{n+1}(t−\mu)^{-n-m-2}$ attains the same value”.
Perhaps other corrections might be needed?
As an extra, could anyone provide a reference to material on how to work out what the ‘Riemann Surface’, for $(t^2−1)^{n+1}(t−\mu)^{-n-m-2}$, actually is, or to material on working out Riemann surfaces in general?
Other:This is prompted by a comment by ”Gateau au fromage”, from 11th Oct 2022.
The closed curve used by Hobson$^1$, starting on pg183, to define the function $\nu$ is a complicated four loop thing. We may take there to be two essentially different loops, each traversed both clock-wise and anticlock-wise. One of the loops encloses the point $t=\mu$, the other encloses the point $t=1$. It is explained on pg 188 that the defining contour integral for ‘Legendre’s Associated Function Of The First Kind $P_n^m(\mu)$’, which is related to the function $\nu$, uses what he calls a ‘Cross-Cut’ from the point $t=+1$ to $t=- \infty$, so the two loopings enclosing $t=1$ used in the definition of $\nu$ should cross this cut-line once.
I guess that this ensures
- that the function
\begin{equation*} \frac { (t^2-1)^{n+1}}{ ~~~(t-\mu)^{n+m+2}} \end{equation*}
returns to the value that it had at the start of the contour, when the contour is completed,
and that
- our function ‘$\nu$’ can have non-zero values.
Reference
1, E.W.Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, 1931.
