Lectures on the Theory of Elliptic Functions

9780486604831: Lectures on the Theory of Elliptic Functions
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ...Possibly the clearest and simplest method of treating this problem is in connection with the Riemann surface upon which the associated integrals may be represented. Before proceeding to the problem of inversion we shall therefore consider this surface in the next Chapter. EXAMPLE 1. If two doubly periodic functions f(z) and jz) have only two poles of the first order in the period-parallelogram and if each pole of the one function coincides with a pole of the other, then is m-cm + c„ where C and C are constants. CHAPTER VI THE RIEMANN SURFACE Article 108. At the close of the preceding Chapter we were left with the discussion of an integral which contained a radical. Such an expression is two-valued, and we must now consider more closely the meaning of such functions and their associated integrals. Take as simplest case the example 8=±Vz-a=±(z-a), where 2 is a complex variable and a an arbitrary constant. For the value z = o, we have s = 0; but for all other finite values of z there are two values of s that are equal and of opposite signs. The point a is called a branch-point of s. The point z = 00 is also a branch-point of this function; for-= = 0 for z = 00. Consequently--and likewise s has s ± V z-a s only one value for z = 00. There are other reasons why z = a and z = 00 are called branchpoints. Corresponding to the value z = zo, let s = s6 l)e a definite value of s. Along the curve (1) from z0 to z consider the values of s at all the points of the curve which differ from one another by infinitesimally small quantities, and similarly consider the values of s along the curve (2) until we again come to z. The value of s at this point will be the same whether we have gone over the first or second curve, provided the branch-point a is not...

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