La Place 8

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Auxiliary Sections > Integral Transforms > Tables of Laplace Transforms > Laplace Transforms: Expressions with Bessel and Modified Bessel Functions

Laplace Transforms: Expressions with Bessel and Modified Bessel Functions Z Original function, f (x)

No

Laplace transform, fe(p) =

∞

0

e−px f (x) dx

1

p

1

J0 (ax)

2

Jν (ax),

3

xn Jn (ax),

n = 1, 2, . . .

¡ ¢−n−1/2 1 ⋅ 3 ⋅ 5 . . . (2n − 1)an p2 + a2

4

xν Jν (ax),

ν > − 12

¡ ¢ ¡ ¢−ν−1/2 2ν π −1/2 Γ ν + 12 aν p2 + a2

5

xν+1 Jν (ax),

6

¡ √ ¢ J0 2 ax

7

√

p2

+ a2

aν ¡ p ¢ν p2 + a2 p + p2 + a2

p

ν > −1

ν > −1

¡ ¢ ¡ ¢−ν−3/2 2ν+1 π −1/2 Γ ν + 32 aν p p2 + a2 1 −a/p e p √ a −a/p e 2 p

¡ √ ¢ xJ1 2 ax

8

¡ √ ¢ xν/2 Jν 2 ax ,

9

¡ √ ¢ J0 a x2 + bx

p

10

I0 (ax)

p

11

Iν (ax),

12

xν Iν (ax),

13

xν+1 Iν (ax),

14

¡ √ ¢ I0 2 ax

1 a/p e p

1 ¡ √ ¢ √ I1 2 ax x ¡ √ ¢ xν/2 Iν 2 ax ,

¢ 1 ¡ √ ea/p − 1 a

15 16

ν > −1

aν/2 p−ν−1 e−a/p 1 p2

+

a2

p ¡ ¢ exp bp − b p2 + a2

1 p2 − a2 aν ¡ p ¢ν p2 − a2 p + p2 − a2

p

ν > −1 ν > − 12 ν > −1

ν > −1

¢ ¡ ¢−ν−1/2 ¡ 2ν π −1/2 Γ ν + 12 aν p2 − a2 ¡ ¢ ¡ ¢−ν−3/2 2ν+1 π −1/2 Γ ν + 32 aν p p2 − a2

aν/2 p−ν−1 ea/p

Z No

Laplace transform, fe(p) =

Original function, f (x)

17

Y0 (ax)

18

K0 (ax)

0

∞

e−px f (x) dx

2 arcsinh(p/a) p π p2 + a2 ¡ p ¢ ln p + p2 − a2 − ln a p p 2 − a2

−

Notation: Jν (z) is the Bessel function of the first kind, Yν (z) is the Bessel function of the second kind, Iν (z) is the modified Bessel function of the first kind, Kν (z) is the modified Bessel function of the second kind. 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 Bessel and Modified Bessel Functions c 2005 Andrei D. Polyanin Copyright °

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