View source for Laplace integral

An integral of the form

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that defines the integral [[Laplace transform|Laplace transform]] of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057490/l0574902.png" /> of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057490/l0574903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057490/l0574904.png" />, giving a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057490/l0574905.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057490/l0574906.png" />. It was considered by P. Laplace at the end of the eighteenth and beginning of the 19th century; it was used by L. Euler in 1737.

Two specific definite integrals depending on the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057490/l0574907.png" />:

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====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Oberhettinger,   L. Badii,   "Tables of Laplace transforms" , Springer  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.N. Sneddon,   "The use of integral transforms" , McGraw-Hill  (1972)  pp. Chapt. 6</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.A. Ditkin,   A.P. Prudnikov,   "Integral transforms" , Plenum  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Doetsch,   "Handbuch der Laplace-Transformation" , '''1–3''' , Birkhäuser  (1950–1956)</TD></TR></table>