Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24

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Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 24

Published $1968$.


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$24 \quad$ Bessel Functions

Bessel's Differential Equation

$24.1$: Bessel's Differential Equation


Bessel Functions of the First Kind of Order $n$

$24.2$: Series Expansion of Bessel Function of the First Kind

\(\displaystyle \map {J_n} x\) \(=\) \(\displaystyle \dfrac {x^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 n + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {n + k + 1} } \paren {\dfrac x 2}^{n + 2 k}\)


$24.3$: Series Expansion of Bessel Function of the First Kind: Negative Index

\(\displaystyle \map {J_{-n} } x\) \(=\) \(\displaystyle \dfrac {x^{-n} } {2^{-n} \, \map \Gamma {1 - n} } \paren {1 - \dfrac {x^2} {2 \paren {2 - 2 n} } + \dfrac {x^4} {2 \times 4 \paren {2 - 2 n} \paren {4 - 2 n} } - \cdots}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {k + 1 - n} } \paren {\dfrac x 2}^{2 k - n}\)


$24.4$: Bessel Function of the First Kind of Negative Integer Order

$\map {J_{-n} } x = \paren {-1}^n \map {J_n} x$


Bessel Functions of the Second Kind of Order $n$

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Bessel Functions of the Second Kind of Order n

General Solution of Bessel's Differential Equation

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/General Solution of Bessel's Differential Equation

Generating Function for $\map {J_n} x$

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Generating Function for J n (x)

Recurrence Formulas for Bessel Functions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Recurrence Formulas for Bessel Functions

Bessel Functions of Order equal to Half an Odd Integer

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Bessel Functions of Order equal to Half an Odd Integer

Hankel Functions of First and Second Kinds of Order $n$

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Hankel Functions of First and Second Kinds of Order n

Bessel's Modified Differential Equation

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Bessel's Modified Differential Equation

Modified Bessel Functions of the First Kind of Order $n$

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Modified Bessel Functions of the First Kind of Order n

Modified Bessel Functions of the Second Kind of Order $n$

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Modified Bessel Functions of the Second Kind of Order n

General Solution of Bessel's Modified Equation

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/General Solution of Bessel's Modified Equation

Generating Function for $\map {I_n} x$

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Generating Function for I n (x)

Recurrence Formulas for Modified Bessel Functions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Recurrence Formulas for Modified Bessel Functions

Modified Bessel Functions of Order equal to Half an Odd Integer

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Modified Bessel Functions of Order equal to Half an Odd Integer

$\operatorname {Ber}$ and $\operatorname {Bei}$ Functions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Ber and Bei Functions

$\operatorname {Ker}$ and $\operatorname {Kei}$ Functions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Ker and Kei Functions

Differential Equation for $\operatorname {Ber}$, $\operatorname {Bei}$, $\operatorname {Ker}$, $\operatorname {Kei}$ Functions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Differential Equation for Ber, Bei, Ker, Kei Functions

Graphs of Bessel Functions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Graphs of Bessel Functions

Indefinite Integrals involving Bessel Functions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Indefinite Integrals involving Bessel Functions

Definite Integrals involving Bessel Functions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Definite Integrals involving Bessel Functions

Integral Representations for Bessel Functions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Integral Representations for Bessel Functions

Asymptotic Expansions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Asymptotic Expansions

Orthogonal Series of Bessel Functions

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Orthogonal Series of Bessel Functions

Miscellaneous Results

Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 24/Miscellaneous Results

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