Special Functions

(1)

(2)

Special Functions

Derivatives, Integrals, Series and Other Formulas

H A N D B O O K O F

(3)

(4)

Yury A. Brychkov

Computing Center of the Russian Academy of Sciences

Moscow, Russia

Special Functions

Derivatives, Integrals, Series and Other Formulas

H A N D B O O K O F

(5)

Chapman & Hall/CRC Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742

© 2008 by Taylor & Francis Group, LLC

Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works

Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1

International Standard Book Number-13: 978-1-58488-956-4 (Hardcover)

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(6)

The Derivatives

1.1. Elementary Functions

!

"

#

# $

%"

&

'

(

&

!

( )

*

1.2. The Hurwitz Zeta Function ζ(ν, z)

^'

+

*

*,

!

"

!

'

+

*

,

!

"

!

"

'

1.3. The Exponential Integral Ei (z)

^&

+

*

,

!

"

!

&

1.4. The Sine si (z) and Cosine ci (z) Integrals

⁽

# +

*

*,

!

"

!

(

1.5. The Error Functions erf (z) and erfc (z)

^-

'

+

*

*,

!

"

!

-

1.6. The Fresnel Integrals S(z) and C(z)

^.

& +

*

,

!

"

!

.

1.7. The Generalized Fresnel Integrals S(z, ν) and C(z, ν)

^.

( +

*

*,

!

"

!

.

1.8. The Incomplete Gamma Functions γ(ν, z) and Γ(ν, z)

-

+

*

*,

!

"

!

-

+

*

*,

!

"

!

"

1.9. The Parabolic Cylinder Function D ν (z)

/

+

*

*,

!

"

!

/

+

*

,

!

"

!

0

#

(7)

1.10. The Bessel Function J ν (z)

^'

.

+

*

*,

!

"

!

'

.

+

*

,

!

"

!

0

-

1.11. The Bessel Function Y ν (z)

^/

+

*

*,

!

"

!

/

+

*

*,

!

"

!

0

1.12. The Hankel Functions H ν ⁽¹⁾ (z) and H ν ⁽²⁾ (z)

+

*

*,

!

"

!

+

*

*,

!

"

!

0

1.13. The Modified Bessel Function I ν (z)

+

*

,

!

"

!

+

*

*,

!

"

!

0

'

1.14. The Macdonald Function K ν (z)

^&

# +

*

*,

!

"

!

&

#

+

*

*,

!

"

!

0

# .

1.15. The Struve Functions H ν (z) and L ν (z)

^#

'

+

*

,

!

"

!

#

'

+

*

*,

!

"

!

0

#

1.16. The Anger J ν (z) and Weber E ν (z) Functions

^#^'

& +

*

*,

!

"

!

# '

&

+

*

*,

!

"

!

0

#(

1.17. The Kelvin Functions ber ν (z), bei ν (z), ker ν (z) and kei ν (z)

^#^-

( +

*

,

!

"

!

# -

(

+

*

*,

!

"

!

0

'

1.18. The Legendre Polynomials P n (z)

^'^&

-

+

*

*,

!

"

!

'

&

1.19. The Chebyshev Polynomials T n (z) and U n (z)

^'^/

/

+

*

,

!

"

!

'/

1.20. The Hermite Polynomials H n (z)

^&

.

+

*

,

!

"

!

&

1.21. The Laguerre Polynomials L ^λ n (z)

^&

+

*

*,

!

"

!

&

+

*

*,

!

"

!

"

&#

1.22. The Gegenbauer Polynomials C n ^λ (z)

^&^#

+

*

,

!

"

!

&#

+

*

*,

!

"

!

"

&&

1.23. The Jacobi Polynomials P n ⁽ ^ρ,σ ⁾ (z)

^&^&

+

*

*,

!

"

!

&&

+

*

*,

!

"

&

/

(8)

1.24. The Complete Elliptic Integrals K (z), E (z) and D (z)

^&^/

# +

*

*,

!

"

!

&

/

1.25. The Legendre Function P ν ^µ (z)

⁽^.

'

+

*

*,

!

"

!

( .

'

+

*

,

!

"

(

1.26. The Kummer Confluent Hypergeometric Function

1 F 1 (a; b; z)

⁽

& +

*

,

!

"

!

(

&

+

*

*,

!

"

( '

1.27. The Tricomi Confluent Hypergeometric Function Ψ(a; b; z)

⁽^&

( +

*

*,

!

"

!

(&

(

+

*

,

!

"

((

1.28. The Whittaker Functions M µ,ν (z) and W µ,ν (z)

^-^.

-

+

*

*,

!

"

!

-.

1.29. The Gauss Hypergeometric Function 2 F 1 (a, b; c; z)

^-^.

/

+

*

,

!

"

!

-.

/

+

*

*,

!

"

-'

1.30. The Generalized Hypergeometric Function p F q ((a p ); (b q ); z)

^-^&

.

+

*

*,

!

"

!

-

&

.

+

*

,

!

"

-

(

Limits

^/^'

2.1. Special Functions

^/^'

!

0

/'

!

*

H

⁰

L

^/^'

!

*

0

/'

#

!

0

"

%

/

&

'

!

%

!

*"

%

0

/

&

!

$

"

%

/

(

!

"

%

/

(

-

!

"

%

/-

/

!

"

%

!

/-

.

$

%

"

//

"

Indefinite Integrals

^.

3.1. Elementary Functions

^.

!

.

3.2. Special Functions

^.

!

#

0

#

.

(9)

!

*

H

⁰

L

^.^'

!

%

0

.'

#

./

Definite Integrals

4.1. Elementary Functions

#

!

"

&

#

$

%"

(

##

# '

!

#& )

*

''

4.2. The Dilogarithm Li 2 (z)

⁽^-

#

)

0

( -

#

)

0

( /

#

)

0

!

-.

#

# )

0

*

-.

4.3. The Sine Si (z) and Cosine ci (z) Integrals

^-

#

)

0

-

#

)

0

-

#

)

0

!

-

#

# )

0

*

-

#

'

)

"

0

-

#

4.4. The Error Functions erf (z), erfi (z) and erfc (z)

^-^#

## )

0

-

#

##

)

0

!

"

-'

##

)

0

-

&

### )

0

!

-

(

## '

)

0

*

-

(

##& )

" 0

0

--

4.5. The Fresnel Integrals S(z) and C(z)

^-^/

# '

)

0

-/

# '

)

0

/.

# '

)

0

!

/

# '

# )

0

/

# '

'

)

0

/

# '

& )

0

!

/

4.6. The Incomplete Gamma Function γ(ν, z)

^/

#& )

0

/

#&

)

0

!

"

/

#&

)

0

/

(10)

#&# )

0

!

/'

#& '

)

0

/'

#&& )

" 0

/'

4.7. The Bessel Function J ν (z)

^/^&

#( )

0

/

&

#(

)

0

!

"

/

(

#(

)

0

/

(

#(# )

0

!

//

#( '

)

0

*

//

#(& )

0

..

#(( )

"

0

.

4.8. The Bessel Function Y ν (z)

^.^#

# -

)

0

.

#

# -

)

0

.

#

4.9. The Modified Bessel Function I ν (z)

^.^'

# /

)

0

.'

# /

)

0

!

"

.

&

# /

)

0

.-

# /

# )

0

!

.

# /

'

)

0

*

# /

& )

0

"

# /

( )

" 0

4.10. The Macdonald Function K ν (z)

^&

# .

)

0

&

# .

)

"

0

&

4.11. The Struve Functions H ν (z) and L ν (z)

⁽

# )

H

L

⁰ ⁽

#

)

H

⁰^!^%^" ^/

#

)

H

L

⁰ ^/

## )

H

L

⁰^!^! ^.

# '

)

H

L

⁰^*

4.12. The Kelvin Functions ber ν (z), bei ν (z), ker ν (z) and kei ν (z)

#

)

0

4.13. The Airy Functions Ai (z) and Bi (z)

#

)

" 0

0

4.14. The Legendre Polynomials P n (z)

## )

0

##

)

0

&

##

)

0

!

Special Functions

Special Functions

Derivatives, Integrals, Series and Other Formulas

H A N D B O O K O F

Yury A. Brychkov

Computing Center of the Russian Academy of Sciences

Moscow, Russia

Special Functions

Derivatives, Integrals, Series and Other Formulas

H A N D B O O K O F

Chapman & Hall/CRC Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742

© 2008 by Taylor & Francis Group, LLC

Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works

Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1

International Standard Book Number-13: 978-1-58488-956-4 (Hardcover)

For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com

Contents

The Derivatives

1.1. Elementary Functions

1.2. The Hurwitz Zeta Function ζ(ν, z)

1.3. The Exponential Integral Ei (z)

1.4. The Sine si (z) and Cosine ci (z) Integrals

1.5. The Error Functions erf (z) and erfc (z)

1.6. The Fresnel Integrals S(z) and C(z)

1.7. The Generalized Fresnel Integrals S(z, ν) and C(z, ν)

1.8. The Incomplete Gamma Functions γ(ν, z) and Γ(ν, z)

1.9. The Parabolic Cylinder Function D ν (z)

1.10. The Bessel Function J ν (z)

1.11. The Bessel Function Y ν (z)

1.12. The Hankel Functions H ν (1) (z) and H ν (2) (z)

1.13. The Modified Bessel Function I ν (z)

1.14. The Macdonald Function K ν (z)

1.15. The Struve Functions H ν (z) and L ν (z)

1.16. The Anger J ν (z) and Weber E ν (z) Functions

1.17. The Kelvin Functions ber ν (z), bei ν (z), ker ν (z) and kei ν (z)

1.18. The Legendre Polynomials P n (z)

1.19. The Chebyshev Polynomials T n (z) and U n (z)

1.20. The Hermite Polynomials H n (z)

1.21. The Laguerre Polynomials L λ n (z)

1.22. The Gegenbauer Polynomials C n λ (z)

1.23. The Jacobi Polynomials P n ( ρ,σ ) (z)

1.24. The Complete Elliptic Integrals K (z), E (z) and D (z)

1.25. The Legendre Function P ν µ (z)

1.26. The Kummer Confluent Hypergeometric Function

1 F 1 (a; b; z)

1.27. The Tricomi Confluent Hypergeometric Function Ψ(a; b; z)

1.28. The Whittaker Functions M µ,ν (z) and W µ,ν (z)

1.29. The Gauss Hypergeometric Function 2 F 1 (a, b; c; z)

1.30. The Generalized Hypergeometric Function p F q ((a p ); (b q ); z)

Limits

2.1. Special Functions

H

L

Indefinite Integrals

3.1. Elementary Functions

3.2. Special Functions

H

L

Definite Integrals

4.1. Elementary Functions

4.2. The Dilogarithm Li 2 (z)

4.3. The Sine Si (z) and Cosine ci (z) Integrals

4.4. The Error Functions erf (z), erfi (z) and erfc (z)

4.5. The Fresnel Integrals S(z) and C(z)

4.6. The Incomplete Gamma Function γ(ν, z)

4.7. The Bessel Function J ν (z)

4.8. The Bessel Function Y ν (z)

4.9. The Modified Bessel Function I ν (z)

4.10. The Macdonald Function K ν (z)

4.11. The Struve Functions H ν (z) and L ν (z)

H

L

H

H

1.12. The Hankel Functions H ν ⁽¹⁾ (z) and H ν ⁽²⁾ (z)

1.21. The Laguerre Polynomials L ^λ n (z)

1.22. The Gegenbauer Polynomials C n ^λ (z)

1.23. The Jacobi Polynomials P n ⁽ ^ρ,σ ⁾ (z)

1.25. The Legendre Function P ν ^µ (z)