Concept of a Riemann Surface by Weyl Hermann; MacLane Gerald R.;

Author:Weyl, Hermann; MacLane, Gerald R.;
Language: eng
Format: epub
Publisher: Dover Publications

If, instead of df one applies the formula (11.8) to the differential df – A dsβ, which has the residue 0 at and , then

from which the congruence (11.8) mod A is turned into an equation. The difference ∫β dsα – ch (β, α) remains unaltered if β is replaced by a path β*joining the same two points and , and therefore will be written in abbreviated form as Naturally, this must be so, for the expression ∫[dsα, df] – ∫αdf contains df but not a path β joining its two singularities and .

With an eye to the function theoretic applications, we supplement equation (11.10) with the following. Let uα be a continuously differentiable function everywhere except at the endpoints and of α, and let duα have infinities of at most the first order at and . Then the equation

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