as Bessel functions

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1: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

2: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

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§35.5(i) Definitions

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§35.5(ii) Properties

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§35.5(iii) Asymptotic Approximations

►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).

3: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

►The function $\varphi (\rho ,\beta ;z)$ is defined by … ►The Laplace transform of $\varphi (\rho ,\beta ;z)$ can be expressed in terms of the Mittag-Leffler function: … ►For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).

4: 10.76 Approximations

§10.76 Approximations

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§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions

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Real Variable and Order $:$ Zeros

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Real Variable; Imaginary Order

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Spherical Bessel Functions

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5: 10.73 Physical Applications

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§10.73(i) Bessel and Modified Bessel Functions

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§10.73(ii) Spherical Bessel Functions

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6: 28.27 Addition Theorems

… ►They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …

7: 10.77 Software

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§10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)

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§10.77(iii) Bessel Functions–Real Order and Argument

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§10.77(vi) Bessel Functions–Imaginary Order and Real Argument

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§10.77(ix) Integrals of Bessel Functions

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§10.77(x) Zeros of Bessel Functions

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8: 10.74 Methods of Computation

… ► ►Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions. … ► … ► ►

§10.74(vii) Integrals

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9: 10.44 Sums

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§10.44(i) Multiplication Theorem

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§10.44(ii) Addition Theorems

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Graf’s and Gegenbauer’s Addition Theorems

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§10.44(iii) Neumann-Type Expansions

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§10.44(iv) Compendia

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10: 10 Bessel Functions

Chapter 10 Bessel Functions

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