## Extract

1. Many important applications of analysis to number-theory require the study of a function *f (s)* of a complex variable *s* = σ + *i*τ near a singular point *s*_{0} = σ_{0} + *i* τ_{0}. The functions *f (s)* is frequently defined for σ > σ_{0} by an infinite series, really d Dirichlet's series, the general term of which is a function of the variables of summation, *e. g*., a quadratic form, raised to the power *s*. Thus the question of finding the number of classes of binary quadratic forms of given determinant, or the number of classes of ideals in a given field, depends upon the residue, Say R, of an appropriate *f (s)* at a simple pole *s*_{0}. A deeper question then suggested is that of finding lim_{s→s0} (*f (s)* — R/*s-s*_{0}). In particular, Kronecker's fundamental formula arises when *f (s)* is a homogeneous binary quadratic form in the variables of summation. Thus, let a *a*(≠ 0), *b, c* be any constants real or complex which are such that the roots ω_{1}, ω_{2} of the quadratic form *ϕ (x, y)* = *ax*^{2} + *bxy* + *cy*^{2} = *a* (*x* - ω_{1}*y*) (*x* - ω_{2}*y*) are neither real nor equal. We need only distinguish the two cases (I) I (ω_{1}) > 0, I (ω_{2}) < 0, (II) I (ω_{1}) > 0, I (ω_{2}) > 0, as the others can be included by writing —*y* for *y*.

## Footnotes

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- Received May 25, 1929.

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