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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: 29. Laplace Transforms 29.1. Definition of the Laplace Transform One-dimensional Laplace Transform 29.1.1 j(s)=-Y{ F ( t ) ) =sm e-"F(t)dt F(t) is a function of the real variable t and s is a complex variable. F(t) is called the original func- tion and j ( s ) is called the image function. If the integral in 29.1.1 converges for a real s=so, i.e., limJAB e-sotF(t)dt A-tO B-t m exists, then it converges for all s with Ws>so, and the image function is a single valued analytic 0 29.2.1 29.2.2 29.2.3 29.2.4 29.2.5 29.2.6 29.2.7 29.2.8 29.2.9 29.2.10 function of s in the half-plane Ws>so. Two-dimensional Laplace Transform 29.1.2 j(u, v ) = Y { F(z, y) } =smsm e-UZ-oYF(x, y ) d d y 0 0 Definition of the Unit Step Function 0 (t Differentiation -tF(t) f ' (8) (-l)"t"F(t) Yn' (8) 1 Adapted by permission from R. V. Churchill, Operational mathematics, 2d ed., McGraw-Hill Book CO., Inc., New York, N.Y., 1958. 1020 The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.