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By W. W. Rouse Ball

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Ty à [a b 0 I c d l/k2 x FIG. 10 Corresponding to abed in the w-plane we get the positive quadrant of the C-plane. Since arg z = 2 arg ζ the positive quadrant of the C-plane becomes the upper half z-plane. It is left as an exercise for the reader to discuss what regions of the z-plane correspond to the rectangle abed in the w-plane by z = en w and z = dn w. The results obtained may be checked by reference to (III), where these transformations are considered in more detail. (III) z = enw and z = dn w.

E. arg(dw/dz) is constant. If dw ^ - = Ciz-af-^z-bf-1... dz (z-fc)*-1, then arg(dw/dz) is constant. If z passes the point a by a small semicircle above it, arg(z — a) decreases from π to 0, the arguments of the other factors remain unchanged, so arg(dw/dz) decreases by π(α—1). Hence the curve in the w-plane turns through the angle π(1 —a) in the positive sense. This corresponds to an angle πα of the polygon. (t-k)*-1 dt, where the constant C may be complex. As |i| -* oo the integrand is 0(l/|i| 2 ) so the integral converges as z -► ± oo, to the same value in each case, since the integral along a large semicircle above the real axis tends to 0.

That, *k z z < r. Since r can be as near as we please to 1, we thus have ^ U l z \ for|z|

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A short account of the history of mathematics by W. W. Rouse Ball


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