# Lectures Introductory to the Theory of Functions of Two Complex Variables; Delivered to the University of Calcutta During January and February 1913

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1914 edition. Excerpt: ...Assign an arbitrary value a' to z' in this restricted domain, that is, such that la' 5'. Then f (z, a') is a function of a single variable only; it is uniform; and it possesses no essential singularity; it is therefore a rational function of z, so that we may write, B, + B, z + + B, z' f" " ' U., Tt', TT+ z, z As a rational function of 2 has a limited number of zeros and of poles, the highest index of z in the numerator and the denominator combined is finite: that is, r is a finite integer. N o generality is lost by assuming that B, and C', are not zero together. If B, were zero, then z= 0 and z' = a' would be a zero of f (z, 2'), contrary to the supposition that f does not vanish within the selected domain; if C', were zero, then z = O and z' = a' would be a pole of f(z, 2'), contrary to the supposition that f is regular within the selected domain; hence neither B, nor C, is zero. Let K0, K1, K2, respectively denote the values of the rational functions A0, A1, A., when z' = a'. Then a converging series forf(z, a') is given by holding for all values of z such that z8. The two coefficients of each power of z on the two sides must be equal to one another; and therefore, as z"" (for n 2 1) does not occur on the right-hand side, we have the coefiicient of z'+" on the left-hand side equal to zero. Thus all the determinants ' K, K, K, ...... i K, K, K, ...... ' ii must vanish. With each value of a', some finite integer r must be associated because f(z, a') is rational in z. But with at least one value (and, it may be, with more than one value) of r, an infinite number of values of a' must be...