an:01380418
Zbl 0932.45008
Golovach, G. P.
Solution of Schl??milch integral equation
UK
Visn., Ser. Fiz.-Mat. Nauky, Ky??v. Univ. Im. Tarasa Shevchenka 1996, No. 2, 7-10 (1996).
00062260
1996
j
45G10 45H05
explicit solution formula; Schl??milch integral equation; continuous solution
The Schl??milch integral equation
\[
f(x) = \frac {2}{\pi}\int _0^{\pi /2}\varphi (x\sin \theta) d\theta
\]
is considered. An effective method of solution of this equation and its generalizations are proposed. It is proved that if a continuous solution \(\varphi \) of the equation \((1)\) exists and the function \(f\) is continuous in the corresponding domain, then the solution of equation (1) is given by the formula
\[
\varphi (t) = \text{sgn}\{t\}\frac {d}{dt}\int_0^t \frac {xf(x)dx}{\sqrt {t^2-x^2}}; \qquad t\not = 0.
\]
The corresponding formula for the equation
\[
f(x) = \frac {2}{\pi}\int _0^{\pi /2} \varphi (x\sin ^\alpha \theta)d\theta, \qquad \alpha >0,
\]
is obtained, too.
O.A.Voina (Ky??v)