# Get A short account of the history of mathematics PDF

By W. W. Rouse Ball

This quantity is made out of electronic photos created in the course of the collage of Michigan collage Library's upkeep reformatting software.

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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|