Elements of the Theory of Functions by Konrad Knopp

Author:Konrad Knopp [Konrad Knopp]
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2016-02-14T16:00:00+00:00

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35The rows of the first matrix are “combined” with the columns of the next; i.e., the sum of the products of the corresponding elements of the two is formed,—just as when multiplying determinants.

36Or , with a ≠ 0.

37If the auxiliary plane of Fig. 17 is used, the individual steps can be followed even more closely.

38The same terminology, of course, is used for entire linear mappings: a translation is said to be parabolic; a rotary stretching with the fixed point as center (cf. (5)) is said to be loxodromic; a pure stretching, hyperbolic; a pure rotation, elliptic.

39One of the points zv, as well as one of the wv, (v = 1, 2, 3), may also be the point ∞.

40Thus, if, e.g., the axis of imaginaries is oriented from bottom to top, the left half-plane is the interior, the right half-plane is the exterior.

41The order in which the four points are taken is not essential, but, of course, once it has been chosen, it must be retained. If the four points are permuted in all possible ways, we do not obtain 24 distinct values of the cross ratio, but, on the contrary, at most 6. If one of the values is equal to δ, the others are 1/δ, 1 – δ, 1/(1 – δ), δ /(δ – 1), and (δ – l)/δ. These values may coincide in part.

42It can be shown that the most general function which maps the interior of one circle in a one-to-one and conformal manner on the interior of another circle, is a linear function. See, e.g., L. R. Ford, Automorphic Functions, New York, 1929, p. 32.

43We purposely order the points in such a manner, that w3 = ∞. For then (w1, w2; w3, w) has the simple form (w – w1)/(w2 – w1), and thus contains the variable w, for which we must finally solve, only in the numerator.

44It is earnestly recommended that the reader make simple sketches for all of the mappings discussed, letting corresponding points and parts of boundaries or regions become clear by using the same colors or hatchings.

45The “base points” ζ1 and ζ2 of the pencil are found by drawing the line of centers of the circles, ζ1 and ζ2 then separate harmonically the pair of points of intersection of each of the circles with this line.

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