Charles Fox

The

The kernels are said to be symmetrical if k(x)=h(x) and unsymmetrical if k(x)9£h(x). The symmetrical case only will concern us here. Various sets of conditions have been discovered which ensure the validity of (1), (2), the set we use here consists of convergence conditions on/(x), k(x) and h(x) together with a functional equation satisfied by the Mellin transforms of k(x) and h(x). K(s) is said to be the Mellin transform of k(x) if

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If a 0 then (1) is a so-called improper integral owing to the infinity in the integrand at x = u. When n = 0 we have associated with (1) the well-known Cauchy principal value, namely (2) . Hadamard (1, p. 117 et seq.) derives from an improper integral an expression which he calls its finite part and which, as he shows, possesses many important properties.

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(M. M. Crum [5], Miss I. Busbridge [l]). In this paper I shall discuss (1) by methods which differ from those used by the three authors cited above. I use (2) to transform (1) into a singular integral equation and this can be solved, in turn, by methods fully described in the standard work on this subject by Muskhelishvili, [6]. Solutions of singular integral equations differ in appearance very considerably from (3a). Singular integral equations occur in many branches of mathematical physics, e.g. elasticity, aerodynamics, etc., and the methods used to solve them are now classical. These solutions lend themselves to numerical computation just as readily as (3a) and it is frequently possible to draw important theoretical and practical conclusions from them. In physical applications of (1) certain restrictions are necessary, such as

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A method is described for finding the frequency functions of bivariate random variables which are the products or ratios of other bivariate random variables. If (ξ, n ) are a pair of bivariate random variables with joint frequency function f ( x, y ) then the method depends upon the fact that the expectation of │ ξ │ r –1 │η│ s –1 is related to the Mellin transform of f ( x, y ) in two dimensions. Knowing the expectation we can then recover the frequency function by means of the inverse Mellin transform. Some examples are given to illustrate the theory.

functions as symmetrical Fourier kernels

The problem discussed is that of solving the integral equation where g ( x ) is given, K v ( z ) is associated with Bessel functions of purely imaginary argument and f ( x ) is to be determined. I prove that, by means of fractional integration, it is possible to reduce this equation to the form of a Laplace transform which can be solved by known methods.

A generalization of the Cauchy principal value

To find two unknownsL, the latitude, and λ, the longitude, two equations are required. The navigator constructs the navigational triangles of two stars and obtains two equations for L and λ. Unfortunately it is almost impossible to solve these equations for L and λ, because of the difficulty of separating the L terms from the λ terms. I consider here the case where the navigator constructs the navigational triangles of three stars and so obtains three equations involving L and λ. If the two time intervals between the measurements of the altitudes of the three stars are small enough one obtains a set of three equations from which L and λ can be easily found with reasonable accuracy. If this accuracy is not good enough another set of three equations can be obtained from which L and λ can be found by the method of Successive Approximations, §9. This method gives good accuracy even if the time intervals are not so small. Many modern calculators can perform arithmetical and trigonometrical operations of great complexity. Some can carry out programs of prescribed operations many times over. Such calculators can perform all the computations required in the methods briefly described above and can relieve the navigator of all the tedium of computation.

A solution of Chandrasekhar’s integral equation

Much is known about power series with integral exponents, and, as in the case of the Theta-functions, about series whose exponents are the squares of the natural numbers. About other types of power series little is known. I prove here that a certain power series whose exponents are the square roots of the natural numbers is a symmetrical Fourier kernel.

Some applications of Mellin transforms to the theory of bivariate statistical distributions

The object of this note is to prove the following results, all of which hold when |a| (2) If r is any positive integer other than zero, the