The Solution of Some Non Linear Integral Equations With Cauchy Kernels

Arthur S Peters

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z-!^oo I ^ z + LJ 1-z The general solution of (4. 9) is G(z) = G^(z) + /l-z^ p(z) where p(z) must satisfy }-^ 21 (4. 11) P'^IO-P'IC) = .
The function p(z) must be taken so that the properties of e^^^V 2 +ijl-z^ match those of This function is analytic for z not on L and it vanishes like c /z as z — > 00, provided c = / (j)(T)dT ^ 0. Furthermore, in accordance L with the assumptions about h^Kx) admitted in the introduction, the behavior of F(z) in the neighborhood of an endpoint a of L is such t...hat £ _^^(a-z)P(z) = 0; and the limit values P"''(C), F"(0 must satisfy a uniform Holder condition. These properties and the condition (4. 11) imply that p(z) must be analytic everywhere and it must vanish at infinity, i. E. , p(z) = 0.
We have now found that , .. Is, . exp {lii! / ±y'^- L Wi-t'^J(t-z) This gives (4. 15) F^(c) = ± (^-iiT^)yf(r7 . Exp|%f. \ .^iilzMi_| L yi-T^(T-a J (4. 14) F-(0 = ± k . I/IIFj/fTTT . Exp \- ^ f AlLlMl. ) ^ L ji-T2(T-aj The solution of (4. 1) is obtained by subtracting (4.

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