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Solution of multiple generalized integral equations of Schlömilch’s type. (Ukrainian) Zbl 0924.45007
The author solves the integral equation $\left({2\over \pi}\right)^{n} \int\limits_{0}^{+\infty}\ldots\int\limits_{0}^{+\infty}\phi(x_1 \cosh^{\alpha_1}\theta_1, x_2\cosh^{\alpha_2}\theta_2,\ldots,x_{n}\cosh^{\alpha_{n}} \theta_{n})d\theta_1\ldots d\theta_{n}=f(x_1,\ldots, x_{n}),$ $$\alpha_{i}>0$$, $$x_{i}>0$$, $$i=1,\dots, n$$, where $$f,\phi$$ are continuous functions; $$f$$ is a known function; $$\phi$$ is an unknown function. The solution of the given integral equation has the form $\phi(s_1,\ldots,\phi_{n})=(-1)^{n}\prod\limits_{i=1}^{n}s_{i} ^{1-1/\alpha_{i}}{\partial^{n}\over\partial s_1\ldots\partial s_{n}}\int\limits_{s_1}^{+\infty}\ldots \int\limits_{s_{n}}^{+\infty}Afdx_1\ldots dx_{n},$ where $A=\prod\limits_{i=1}^{n}x_{i}^{2/\alpha_{i}-1}/\prod\limits_{i= 1}^{n} (x_{i}^{2/\alpha_{i}}-s_{i}^{2/\alpha_{i}})^{1/2}, x_{i}>s_{i}.$
##### MSC:
 4.5e+11 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)