By Henry Thomas Herbert Piaggio

Initially released in 1920. This quantity from the Cornell college Library's print collections used to be scanned on an APT BookScan and switched over to JPG 2000 layout via Kirtas applied sciences. All titles scanned disguise to hide and pages might comprise marks notations and different marginalia found in the unique quantity.

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**Example text**

Will be assumed to lie in S+. As it is easily seen, it is sufficient to consider the case in which at least one of these points, say lit, lies at a distance from the boundary which is less than some constant e < R o' where Ro is some standard radius. 2) necessarily holds, since 0(z) is holomorphic. Temporarily denote lit by Zo and Zs by z. 6) which is holomorphic in the cut region and continuous fro" 1 the left on its entire boundary. 5), the limiting value f this function satisfies IlJIo+(t) I ;;;;;; C on L.

For example I. I. ) cp(t) will be called the density function. Below (§ 12) a definite meaning will be given to this integral in certain cases where the point z lies on L (but not at one of its ends); meanwhile it will be assumed that the point z does not lie on L. It is clear that the function (1)(z) is holomorphic in the entire region excluding L and that for large I z I (1)(z) = (since L is of finite length). 0(_1 ) Iz I § 11. Connection with logarithmic potential. Historical re- marks. Cauchy integrals are closely related to the potentials of simple and double layers, distributed along the line of integration L.

I ep(t)(t -lI)1 - ep(to) tit, 2m I = I then. since I d8 I ~ K I dr I, I ep(e) - ep(to) I ~ A I , - to I", I '-lIll ~ (r-6coscoo)1 +6l sinl coOo where COo = {Jo-rxo. §18). III~AXf - 2n (r - "'Idrl ~ 6 cos COO)I + 61 sinl COo - lib f ~n AX Re o ",dr· (r - 6 cos COO)I + 61 sinl coo' On writing r = 6. ' f < 1. one obtains GO 1 I I I < AX :rr {)II- ( fllld" " - cos COo )1 + sm. I COo - '---1 Cv• o where C is a constant. ' = 1 the bound for I I I can be evaluated in finite form and reduced to the estimate I I I ~ C I log 6 I.