1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation

. The element of arc of 'Y at p is given by

ds^2 = dx · dx = L(dxi)^2

i

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the dxi being direction numbers of the tangent line to -y* at p. The

corresponding element of arc of r at f(p) is given by

da^2 = dy · dy = L(dyi)^2 = Mjkdxidxk

i

j' k = 1, ... ' n, the dyi being direction numbers of the tangent line to r*

at f(p), and

Hence the square of the magnification ratio p = da / ds is given by

2 Mjkdxidxk

p = I:;( dxi)2

This ratio is independent of the direction numbers dxi iff

for j =/:-k and

M denoting a positive constant at p. Then we have

p2 =M

(6.24-17)

(6~24-18)

With the notation aii = ofi/oxi, and using (6.24-18), equations (6.24-

17. can be written as

(a11)2 + ... + (an1)2 = ... = (aln)2 + .. , + (ann)2 = P2

ana12 + .. , + anlan2 = O

alla13 + ... + anlan3 = O

Consider the n equations containing a^1 i , ... ,ani, namely,

allalj + a21a2j + ... + anlanj = O

alj alj + a2i a2j + .. , + anj anj = P2