solution by variation of parameters

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1: 1.13 Differential Equations

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Variation of Parameters

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2: 9.12 Scorer Functions

… ►Solutions of this equation are the Scorer functions and can be found by the method of variation of parameters (§1.13(iii)). …

3: 10.15 Derivatives with Respect to Order

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10.15.3 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{\pi }{2}{Y}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{J}_k\left(z\right)}{k!(n-k)}.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.66

Referenced by:

§10.15, §10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.4 $${\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=-\frac{\pi }{2}{J}_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{Y}_k\left(z\right)}{k!(n-k)},$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.67

Permalink:

http://dlmf.nist.gov/10.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.5 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=\frac{\pi }{2}{Y}_0\left(z\right),{\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=-\frac{\pi }{2}{J}_0\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.68

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.6 $${\frac{\partial J_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\sin x-\mathrm{Si}\left(2x\right)\cos x\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.41 (corrected)

Referenced by:

§10.15, §10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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10.15.7 $${\frac{\partial J_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =-\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}\left(2x\right)\cos x+\mathrm{Si}\left(2x\right)\sin x\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{Ci}\left(z\right)$: cosine integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.1.42 (corrected)

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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4: 2.8 Differential Equations with a Parameter

§2.8 Differential Equations with a Parameter

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§2.8(i) Classification of Cases

… ►Zeros of $f(z)$ are also called turning points. … ►

§2.8(vi) Coalescing Transition Points

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5: Bibliography M

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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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External Links:

ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

Encodings:

BibTeX

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Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

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External Links:

ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.8(vi).

Encodings:

BibTeX

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Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.

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External Links:

ISSN 0025-5831, Document, MathReview, ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

BibTeX

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Latent roots and matrix variates: A review of some asymptotic results. Ann. Statist. 6 (1), pp. 5–33.

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External Links:

FullText, MathReview (Yasuko Chikuse), ZentralBlatt

Referenced by:

§35.6(i), §35.6(iii).

Encodings:

BibTeX

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The variation with respect to order of zeros of Bessel functions. Rend. Sem. Mat. Univ. Politec. Torino 39 (2), pp. 15–25.

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External Links:

MathReview (S. Ahmed), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

BibTeX

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