La Place 5

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Auxiliary Sections > Integral Transforms > Tables of Laplace Transforms > Laplace Transforms: Expressions with Logarithmic Functions

Laplace Transforms: Expressions with Logarithmic Functions Z Laplace transform, fe(p) =

Original function, f (x)

No 1

ln x

1 − (ln p + C) p

2

ln(1 + ax)

1 − ep/a Ei(−p/a) p

3

ln(x + a)

¤ 1£ ln a − eap Ei(−ap) p

4

xn ln x,

5

1 √ ln x x

n = 1, 2, . . .

n! ¡ 1+ pn+1 −

1 2

1 3

+

+ ··· +

− ln p − C

¢

£ ¤ π/p ln(4p) + C

¤ 2 + · · · + 2n−1 − ln(4p) − C , √ π kn = 1 ⋅ 3 ⋅ 5 . . . (2n − 1) n 2 £ ¤ −ν Γ(ν)p ψ(ν) − ln p kn n+1/2 p

£ 2+

2 3

+

2 5

xn−1/2 ln x,

7

xν−1 ln x,

8

(ln x)2

¤ 1£ (ln x + C)2 + 16 π 2 p

9

e−ax ln x

−

ν>0

e−px f (x) dx

p

6

n = 1, 2, . . .

1 n

0

∞

ln(p + a) + C p+a

Notation: C = 0.5772 . . . is the Euler constant, Ei(z) is the integral exponent, Γ(ν) is the gamma function, ψ(ν) is the logarithmic derivative of the gamma function. References Bateman, H. and Erd´elyi, A., Tables of Integral Transforms. Vols. 1 and 2, McGraw-Hill Book Co., New York, 1954. Doetsch, G., Einf¨uhrung in Theorie und Anwendung der Laplace-Transformation, Birkh¨auser Verlag, Basel–Stuttgart, 1958. Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965. Polyanin, A. D. and Manzhirov, A. V., Handbook of Integral Equations , CRC Press, Boca Raton, 1998.

Laplace Transforms: Expressions with Logarithmic Functions c 2005 Andrei D. Polyanin Copyright °

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