. On Cauchy's modulus surfaces. in which for the second parenthesis reduces t;o a©* Writingz - z^ = u+iv and z - /^^s:^ t there is (9) u-viv -p\ cos ke-vi sin k») ,^(/^, in which for z = z^reduces to some constant, say^t In the neighborhood of z= zo we may therefore writeu-viv ~fof^ ^^s (j^e+Bo )-f-i sin(ke-ve^^, and hence, (lO^ u =/^/^*^cos(ke+eo)V -foP ^sinCke+eo)To any direction v/gi thru z^, correspond in the z- plane the directions defined hy the relationtan (ke+Bj, ) =/i. If is the smallest angle for which tan -/^, we may write , so that Rendiconti del Circolo Matematice Di Palermo.
. On Cauchy's modulus surfaces. in which for the second parenthesis reduces t;o a©* Writingz - z^ = u+iv and z - /^^s:^ t there is (9) u-viv -p\ cos ke-vi sin k») ,^(/^, in which for z = z^reduces to some constant, say^t In the neighborhood of z= zo we may therefore writeu-viv ~fof^ ^^s (j^e+Bo )-f-i sin(ke-ve^^, and hence, (lO^ u =/^/^*^cos(ke+eo)V -foP ^sinCke+eo)To any direction v/gi thru z^, correspond in the z- plane the directions defined hy the relationtan (ke+Bj, ) =/i. If is the smallest angle for which tan -/^, we may write , so that Rendiconti del Circolo Matematice Di Palermo. Iomo xxxiv,AnT^o 193?, On Conformal Rational Transf in n 1 1 ^a^^^d^- i 0,1,fl, , k-1. Any two consecutive branches intersect at an angleTT/k. For a direc-tBon v/u = -1/^ f perpenditcular to the first,cot (ke f &f,) = (k» f e^VTT/^ ) = f 7f/o) ATT e^^-Bo- 7^ «TV^k ; 7V= 0,1,2, (k-1). the corresponding branches in the xy plane hiseCt the anglesformed by the branches of the first 6 Investigation of Simple Cases. I, W = az . 4- t». The transformation W = az V h is a linear transformation, con-sisting of similitude, a rotation, and a translation, a is a complexnumber and may he expressed as Ae^ z ^e^. i\ W = A^.f^\^ - Af^e^*^^^ = Similitude ( co-efficient a) f rotation thru the angle o<,.oncauchysmodulus00grim
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