The object of this note is to give simpler proof* of two formulae involving the function ψ (z) which I have proved elsewhere by more complicated methods.

The series

in which r is zero or an integer is rapidly convergent if x is large but may be very slowly convergent if x is small. The object of this note is to derive an alternative series for S2r(x) which is rapidly convergent for small values of x.

The purpose of this note is to generalise the Dirichlet-Liouville formula which expresses a certain type of multiple integral in terms of a single integral. In our formula the multiple integral will involve several arbitrary functions instead of only one, and it will be expressed as a product of single integrals.

In this note an explicit expression is obtained for the sum of the r-th powers of the first n integers. The result is equivalent to the well-known result in terms of Bernoulli numbers and the equivalence is not difficult to establish. However, the method given here is elementary and self-contained and provides an excellent exercise on. the manipulation of determinants.

Consider the nth order differential equation

where the coefficients cv are real constants and f is a real function continuous in the interval a≦ x ≦ b. The following theorem will be proved in §4:

If the characteristic equation of (I) has no purely imaginary roots, then a particular integral η (x) can always be found which satisfies the inequality

where C is a certain function of the cv only and M is the maximum of |f|. In particular we may take C = 1 if all roots of the characteristic equation are real.