# Article

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Keywords:
integral equations; integral transforms
Summary:
This paper originates in the author's investigations on thermal and mechanical processes in semi-infinite continua. It presents a contribution to the theory of the Laplace transformation by deriving the original functions to the transforms $\frac{e^{-a\sqrt{(p+b^2)}}}{p\sqrt{(p+b^2)}},\ a\geq 0,\ b\neq 0$, $\frac{e^{-a\sqrt{(p^2+b^2)}}}{p^2\sqrt{(p^2+b^2)}},\ a>0,\ b\neq 0$. The significance and, may be, originality of the paper lies in that it is not easy to find reliable results in the existing literature, secondly in that the procedure works quickly and can be extended to deriving other useful operational formulae and finally in that there exist vast groups of physical and technical problems which are closely connected with the results given in the paper. Some examples concerning the heating of conducting semi-space by a variable heat flux through the surface are discussed in considerable details. They can serve to technicians and physicists as a kind of guide in treating other advanced questions of mathematical physics and egineering.
References:
[1] Диткин В. А,, Кузнецов П. И.: Справочник по операционному исчислению. ГИТТЛ, Москва-Ленинград 1951, стр. 159. Zbl 1165.94313
[2] Теумин И. И.: Справочник по переходным электрическим процессам. Связиздат, Москва 1951, стр. 369. Zbl 1165.94313
[3] Carslaw H. S., Jaeger J. С.: Conduction of Heat in Solids. Oxford 1959, str. 264.
[4] Vodička V.: O podélném kmitání polonekonečných tyčí. Čs. Čas. Fys. 13 (1963), 81.

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