# Bessel equation

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## 21—30 of 101 matching pages

##### 21: 10.40 Asymptotic Expansions for Large Argument

###### §10.40 Asymptotic Expansions for Large Argument

►###### §10.40(i) Hankel’s Expansions

… ►###### Products

… ► … ►###### §10.40(iv) Exponentially-Improved Expansions

…##### 22: 9.13 Generalized Airy Functions

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►and ${\mathcal{Z}}_{p}$ is any linear combination of the modified Bessel functions ${I}_{p}$ and ${\mathrm{e}}^{p\pi \mathrm{i}}{K}_{p}$ (§10.25(ii)).
►Swanson and Headley (1967) define independent solutions ${A}_{n}\left(z\right)$ and ${B}_{n}\left(z\right)$ of (9.13.1) by
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$$-{A}_{n}^{\prime}\left(0\right)={p}^{1/2}{B}_{n}^{\prime}\left(0\right)=\frac{{p}^{p}}{\mathrm{\Gamma}\left(p\right)}.$$

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##### 23: 32.10 Special Function Solutions

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###### §32.10(iii) Third Painlevé Equation

…##### 24: Frank W. J. Olver

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►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history.
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►He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i.
…, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e.
…, Bessel functions, hypergeometric functions, Legendre functions).
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##### 25: 10.68 Modulus and Phase Functions

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►Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta $ replaced throughout by $\mathrm{ker}$, $\mathrm{kei}$, $N$, and $\varphi $, respectively.
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►However, numerical tabulations show that if the second of these equations applies and ${\varphi}_{1}\left(x\right)$ is continuous, then ${\varphi}_{1}\left(0\right)=-\frac{3}{4}\pi $; compare Abramowitz and Stegun (1964, p. 433).

##### 26: 30.10 Series and Integrals

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##### 27: 30.2 Differential Equations

##### 28: 10.16 Relations to Other Functions

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###### Elementary Functions

… ►###### Parabolic Cylinder Functions

… ►Principal values on each side of these equations correspond. ►###### Confluent Hypergeometric Functions

… ►###### Generalized Hypergeometric Functions

…##### 29: Bibliography D

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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions.
Proc. Roy. Soc. London Ser. A 440, pp. 37–54.
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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter.
SIAM J. Math. Anal. 21 (4), pp. 995–1018.
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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions.
Stud. Appl. Math. 107 (3), pp. 293–323.
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##### 30: 2.8 Differential Equations with a Parameter

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