By Dr. H. Durege

Writer: G. E. Fisher and that i. J. Schwatt booklet date: 1896 Notes: this can be an OCR reprint. there is various typos or lacking textual content. There are not any illustrations or indexes. if you happen to purchase the overall Books variation of this ebook 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 also preview the publication there.

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**Extra info for Elements of the Theory of Functions of a Complex Variable**

**Example text**

MULTIFORM FUNCTIONS. 47 round each branch-point and let the variable describe the same in succession, then this path can be deformed, without line passing over one of the branchpoints, into a closed line which, starting from branch-points. a and z , encloses all the where two branch- (Fig. ) lines round the individual branchpoints most simply, by describing round each one a small circle, and connecting each of these circles with must then be described 10. We twice, going will next illustrate by some examples, and at the % by a and coming.

Since in this case x is constant and can be easily obtained, by substituting the value of a? in the equations equal to 1 ? 2 z=z of y and v 2xy and eliminating y\ thereby we obtain the equation v 2 4(1 u) between the co-ordinates u and # of the point w, which shows that the locus of w is a parabola, = u = with its vertex at eter, the 1, ordinate its at focus at the focus, o, is and tangents at the points of intersection respond to c and d, it is c which the paramBy examining the and d which cor- of 2.

1. If we let the variable start In this z = is a branch-point. from the point # = 1 and describe the circumference of a circle round the origin, this is a closed line which encloses the branch-point. If the function w = V# start from the point 2 = 1 with the value w = -+- 1, and if we put = r (cos + i sin <), = 0. z = 1, r = 1 and z 4> If z next describe then at the point 4> of increasing in the of circle direction the the circumference increases and to constant and r remains 1, equal angles, If therefore the variable return to the point to 2 TT.