# Elements of the Theory of Functions of a Complex Variable by Dr. H. Durege

By Dr. H. Durege

Writer: G. E. Fisher and that i. J. Schwatt ebook date: 1896 Notes: this is often an OCR reprint. there is a number of typos or lacking textual content. There aren't any illustrations or indexes. should you purchase the overall Books variation of this booklet you get loose trial entry to Million-Books.com the place you could make a choice from greater than 1000000 books at no cost. you may as well preview the e-book there.

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The angles z zz the and included sides are But since this other, proportional. must hold for any pair of corresponding points z and w, the each other; f figure described by the point w is in its infinitesimal elements similar to that described by the point z? and two intersecting curves in the to-plane form with each other the same angle as by the corresponding curves in the #-plane. In that formed this connection it must be noticed that is dz 1 We note that, even if az supposed to be be independent of \$#, yet dw, which = in general changes its direction and magnitude with dz.

W = z2 The simple function may serve as an example. We w = x2 obtain here u=x y 2 and hence + 2 ixy, 2 2 y y v 2xy, Bx _, fy , By which verify the equations of condition (2). /-axis, so that x = 0, then z = iy and w = _ y* j hence w describes the negative part of the principal and only this, so that, when z goes from a through o to &, w moves from a' to o and then back again to 6'; a and &' coincide when ad is assumed equal to ob (Fig. 8). Let 2 further describe a circle with radius r round the origin, so that axis f } when z r remains constant ; = r(cos ^ + i sin <), then w == r^cos 2 *Cf.

Since in , this process the real variables x and y can each vary quite independently of the other, the point representing z can also describe an arbitrary line. , for the quite arbitrary relation in which x and y must stand to each other in every position of the point to be always expressible by the same equation (or, indeed, by any equation whatever). In order that the variation of z may be continuous, it is necessary only for the line to form a continuous trace. A few examples may % G make this clear.