The formula to be proved is

The formulae required in the proof are the Barnes Integral

and Whipple's Formula (1)

§ 2. Proof of the Formula. From (2) the E-function on the left of (1) is equal to

1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we have

The earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams [2]. The proof now to be given follows different lines from his.

The problem of solving the equation of thermal conduction for cases in which heat is generated in the interior of the medium under consideration arises frequently in physics and engineering. It occurs, for instance, when we consider the diffusion of heat in a solid undergoing radioactive decay (1) or which is absorbing radiation (2). Complications of a similar nature arise when there is a generation or absorption of heat in the solid as a result of a chemical change-for example, the hydration of cement (3). The particular case in which the rate of generation of heat is independent of the temperature arises in the theory of the ripening of apples and has been discussed by Awberry (4).

In this note we derive some integrals involving confluent hypergeometric functions and analogous to Lommel's integrals for Bessel functions. Although the method of derivation is straightforward, the integrals do not seem to be mentioned in the literature.

In Muir's Theory of Determinants, Vol. III, pp. 232–237, there will be found accounts of papers by H. Nägelsbach, J. Hammond and J. W. L. Glaisher, in which expressions for the Bernoulli numbers are obtained in terms of determinants. In the present paper an expression for Bn will be derived which appears to be new, but which is very like some of those mentioned by Muir.

The formula to be established is

where l, m, n. are any numbers real or complex and R(b)>0. A similar result, involving Bessel Functions of the First Kind, was obtained by Hanumanta Rao [Mess, of Maths., XLVII. (1918), pp. 134–137].

§ 1. Introductory. The formula to be proved is

where b>0.

This paper describes a simple method of evaluating integrals of a type some cases of which have been discussed by Dr. Ragab.

The following formulae are required in the proofs

If p≧q + 1, z ≠ 0,

If. p = q + 1, | z |>1 in this formula.

By applying Gauss's Theorem it can be seen that, if n is a positive integer and α is not integral,

so that

In section 2 it will be proved that, if

Where α and β are not integers

Where μ and M are constants independent of n.

In § 2 the following two formulae will be established.

If, when p≧q + l, R(ar + k)>0, r = 1, 2,…, p and | ampz |<π,

If p≧q, the result holds if the integral is convergent.

The formulae

where γ = ½ (α + β + l), and

where γ = 1 + β, were given by Gauss (Ges. Werke, iii, pp. 225, 226). It is here proposed to find the corresponding expressions for the hypergeometric function when γ has general values (not zero or negative integral). These will be derived in section 2 by applying Lagrange's expansion

where

and that root of equation (4) in x is taken which is equal to λ when w = 0. Two generalisations of Whipple's Transformation will be obtained in section 3.

1. There are exceptional integrals of the total differential equation

in the case when it is not completely integrable, and so when the invariant

is not identically zero, which do not seem to be mentioned by any standard authorities such as Cartan, Goursat, de la Vallée Poussin, and Schouten and Kulk. These are integrals of (1) which do not reduce I to zero. They arise only when the first partial derivates of P, Q, R are not all continuous. A simple example is z = 0 as an integral of

§ 1. Introduction. A problem of some interest in mathematical statistics is that of determining conditions under which the randomisation distribution of a statistic and its normal theory distribution are asymptotically equivalent, as these two distributions are used in alternative approaches to the same inference problem.

We consider a volume of material, divided into two regions 1 and 2. each of density ρ, by a moving surface S. On S a change of phase occurs, at a definite temperature (which we may take to be zero) and with absorption or liberation of a latent heat L per unit mass. If θl, kl, K1 are the temperature, thermal conductivity and diffusivity of phase 1, and θ2, k2, K2 corresponding quantities for phase 2, the surface S is the isothermal

and the boundary condition on this surface is

Subscript letters denote partial differentiation.

Let f(x, y, z, t) satisfy the equation

For certain purposes, particularly in connection with the propagation of a boundary of fusion etc., it is of interest to discover solutions of (1) which permit the equation:

to be solved explicitly in the form:

This suggests the examination of solutions of the type

where

and f, ϕ, ψ are functions to be determined. To save repetition, Roman capitals denote arbitrary constants throughout.