10 Bessel Functions Spherical Bessel Functions 10.48 Graphs 10.50 Wronskians and Cross-Products

§10.49 Explicit Formulas

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Permalink:: http://dlmf.nist.gov/10.49
See also:: Annotations for Ch.10

§10.49(i) Unmodified Functions
§10.49(ii) Modified Functions
§10.49(iii) Rayleigh’s Formulas
§10.49(iv) Sums or Differences of Squares

§10.49(i) Unmodified Functions

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Keywords:: explicit formulas, spherical Bessel functions, unmodified functions
Notes:: When $\nu=n+\tfrac{1}{2}$ the asymptotic expansions (10.17.3)–(10.17.6) terminate, and as a consequence of the error bounds of §10.17(iv) they represent the left-hand sides exactly.
Referenced by:: §10.16, §10.49(ii), §10.61(v), §10.74(i), §11.4(i)
Permalink:: http://dlmf.nist.gov/10.49.i
See also:: Annotations for §10.49 and Ch.10

Define $a_{k}(\nu)$ as in (10.17.1). Then

10.49.1 $a_{k}(n+\tfrac{1}{2})=\begin{cases}\dfrac{(n+k)!}{2^{k}k!(n-k)!},&k=0,1,\dotsc% ,n,\\ 0,&k=n+1,n+2,\dotsc.\end{cases}$

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Symbols:: : factorial (as in ), : integer, : nonnegative integer and $a_{k}(\nu)$ : polynomial coefficient
Referenced by:: §10.49(ii), §10.50
Permalink:: http://dlmf.nist.gov/10.49.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(i), §10.49 and Ch.10

10.49.2 $\mathsf{j}_{n}\left(z\right)=\sin\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{% \left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+% \cos\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right% \rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\left\lfloor\NVar{x}\right\rfloor$ : floor of , $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 10.1.8
Referenced by:: §10.50, §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(i), §10.49 and Ch.10

10.49.3

$\mathsf{j}_{0}\left(z\right)=\frac{\sin z}{z},$

$\mathsf{j}_{1}\left(z\right)=\frac{\sin z}{z^{2}}-\frac{\cos z}{z},$

$\mathsf{j}_{2}\left(z\right)=\left(-\frac{1}{z}+\frac{3}{z^{3}}\right)\sin z-% \frac{3}{z^{2}}\cos z.$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind and : complex variable
A&S Ref:: 10.1.11 (corrected)
Referenced by:: §10.49(iii), §10.56
Permalink:: http://dlmf.nist.gov/10.49.E3
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §10.49(i), §10.49 and Ch.10

10.49.4 $\mathsf{y}_{n}\left(z\right)=-\cos\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{% \left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+% \sin\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right% \rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\left\lfloor\NVar{x}\right\rfloor$ : floor of , $\sin\NVar{z}$ : sine function, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 10.1.9
Referenced by:: §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(i), §10.49 and Ch.10

10.49.5

$\mathsf{y}_{0}\left(z\right)=-\frac{\cos z}{z},$

$\mathsf{y}_{1}\left(z\right)=-\frac{\cos z}{z^{2}}-\frac{\sin z}{z},$

$\mathsf{y}_{2}\left(z\right)=\left(\frac{1}{z}-\frac{3}{z^{3}}\right)\cos z-% \frac{3}{z^{2}}\sin z.$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind and : complex variable
A&S Ref:: 10.1.12 (corrected)
Referenced by:: §10.50, §10.56
Permalink:: http://dlmf.nist.gov/10.49.E5
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §10.49(i), §10.49 and Ch.10

10.49.6 ${\mathsf{h}^{(1)}_{n}}\left(z\right)=e^{iz}\sum_{k=0}^{n}i^{k-n-1}\frac{a_{k}(% n+\frac{1}{2})}{z^{k+1}},$

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, ${\mathsf{h}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 10.1.16
Referenced by:: §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(i), §10.49 and Ch.10

10.49.7 ${\mathsf{h}^{(2)}_{n}}\left(z\right)=e^{-iz}\sum_{k=0}^{n}(-i)^{k-n-1}\frac{a_% {k}(n+\frac{1}{2})}{z^{k+1}}.$

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Symbols:: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, ${\mathsf{h}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 10.1.17 (corrected)
Permalink:: http://dlmf.nist.gov/10.49.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(i), §10.49 and Ch.10

§10.49(ii) Modified Functions

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Keywords:: Bessel polynomials, explicit formulas, modified functions, spherical Bessel functions
Notes:: Use the same method as for §10.49(i), or combine the results of §10.49(i) with (10.47.12) and (10.47.13).
Referenced by:: §10.39, §10.74(i), §11.4(i), §18.34(i), §18.34(iii), §3.5(viii)
Permalink:: http://dlmf.nist.gov/10.49.ii
See also:: Annotations for §10.49 and Ch.10

Again, with $a_{k}(n+\tfrac{1}{2})$ as in (10.49.1),

10.49.8 ${\mathsf{i}^{(1)}_{n}}\left(z\right)=\tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}% \frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n+1}\*\tfrac{1}{2}e^{-z}\sum_{k=0}^% {n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}.$

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Symbols:: $\mathrm{e}$ : base of natural logarithm, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 10.2.9 (modified)
Referenced by:: §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(ii), §10.49 and Ch.10

10.49.9

${\mathsf{i}^{(1)}_{0}}\left(z\right)=\frac{\sinh z}{z},$

${\mathsf{i}^{(1)}_{1}}\left(z\right)=-\frac{\sinh z}{z^{2}}+\frac{\cosh z}{z},$

${\mathsf{i}^{(1)}_{2}}\left(z\right)=\left(\frac{1}{z}+\frac{3}{z^{3}}\right)% \sinh z-\frac{3}{z^{2}}\cosh z.$

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Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function and : complex variable
A&S Ref:: 10.2.13
Referenced by:: §18.34(i)
Permalink:: http://dlmf.nist.gov/10.49.E9
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §10.49(ii), §10.49 and Ch.10

10.49.10 ${\mathsf{i}^{(2)}_{n}}\left(z\right)=\tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}% \frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n}\tfrac{1}{2}e^{-z}\sum_{k=0}^{n}% \frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}.$

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Symbols:: $\mathrm{e}$ : base of natural logarithm, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 10.2.10 (modified)
Referenced by:: §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(ii), §10.49 and Ch.10

10.49.11

${\mathsf{i}^{(2)}_{0}}\left(z\right)=\frac{\cosh z}{z},$

${\mathsf{i}^{(2)}_{1}}\left(z\right)=-\frac{\cosh z}{z^{2}}+\frac{\sinh z}{z},$

${\mathsf{i}^{(2)}_{2}}\left(z\right)=\left(\frac{1}{z}+\frac{3}{z^{3}}\right)% \cosh z-\frac{3}{z^{2}}\sinh z.$

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Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function and : complex variable
A&S Ref:: 10.2.14
Permalink:: http://dlmf.nist.gov/10.49.E11
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §10.49(ii), §10.49 and Ch.10

10.49.12 $\mathsf{k}_{n}\left(z\right)=\tfrac{1}{2}\pi e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n% +\frac{1}{2})}{z^{k+1}}.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, : integer, : nonnegative integer, : complex variable and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 10.2.15
Referenced by:: §10.52(ii)
Permalink:: http://dlmf.nist.gov/10.49.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(ii), §10.49 and Ch.10

10.49.13

$\mathsf{k}_{0}\left(z\right)=\tfrac{1}{2}\pi\frac{e^{-z}}{z},$

$\mathsf{k}_{1}\left(z\right)=\tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{1}{% z^{2}}\right),$

$\mathsf{k}_{2}\left(z\right)=\tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{3}{% z^{2}}+\frac{3}{z^{3}}\right).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function and : complex variable
A&S Ref:: 10.2.17
Permalink:: http://dlmf.nist.gov/10.49.E13
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §10.49(ii), §10.49 and Ch.10

$\sum_{k=0}^{n}a_{k}(n+\tfrac{1}{2})z^{n-k}$ is sometimes called the Bessel polynomial of degree . For a survey of properties of these polynomials and their generalizations see Grosswald (1978). See also §18.34, de Bruin et al. (1981a, b), and Dunster (2001c).

§10.49(iii) Rayleigh’s Formulas

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Keywords:: Rayleigh’s formulas, spherical Bessel functions
Notes:: For the first of(10.49.14) combine the second of (10.51.3), with and , and the first of (10.49.3). Similarly for the other results.
Permalink:: http://dlmf.nist.gov/10.49.iii
See also:: Annotations for §10.49 and Ch.10

10.49.14

$\mathsf{j}_{n}\left(z\right)=z^{n}\left(-\frac{1}{z}\frac{\mathrm{d}}{\mathrm{% d}z}\right)^{n}\frac{\sin z}{z},$

$\mathsf{y}_{n}\left(z\right)=-z^{n}\left(-\frac{1}{z}\frac{\mathrm{d}}{\mathrm% {d}z}\right)^{n}\frac{\cos z}{z}.$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, : integer and : complex variable
A&S Ref:: 10.1.25, 10.1.26
Referenced by:: §10.49(iii)
Permalink:: http://dlmf.nist.gov/10.49.E14
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.49(iii), §10.49 and Ch.10

10.49.15

${\mathsf{i}^{(1)}_{n}}\left(z\right)=z^{n}\left(\frac{1}{z}\frac{\mathrm{d}}{% \mathrm{d}z}\right)^{n}\frac{\sinh z}{z},$

${\mathsf{i}^{(2)}_{n}}\left(z\right)=z^{n}\left(\frac{1}{z}\frac{\mathrm{d}}{% \mathrm{d}z}\right)^{n}\frac{\cosh z}{z}.$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, : integer and : complex variable
A&S Ref:: 10.2.24, 10.2 25
Permalink:: http://dlmf.nist.gov/10.49.E15
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.49(iii), §10.49 and Ch.10

10.49.16 $\mathsf{k}_{n}\left(z\right)=(-1)^{n}\tfrac{1}{2}\pi z^{n}\left(\frac{1}{z}% \frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}\frac{e^{-z}}{z}.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.49.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(iii), §10.49 and Ch.10

§10.49(iv) Sums or Differences of Squares

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Keywords:: explicit formulas, spherical Bessel functions, sums or differences of squares
Notes:: From (10.18.6), (10.47.3), and (10.47.4), ${\mathsf{j}_{n}}^{2}\left(z\right)+{\mathsf{y}_{n}}^{2}\left(z\right)=(\pi/(2z% )){M_{n+\frac{1}{2}}}^{2}\left(z\right)$ . Then apply (10.18.17). To derive (10.49.20) combine (10.47.12) and (10.49.18).
Permalink:: http://dlmf.nist.gov/10.49.iv
See also:: Annotations for §10.49 and Ch.10

Denote

10.49.17 $s_{k}(n+\tfrac{1}{2})=\frac{(2k)!(n+k)!}{2^{2k}(k!)^{2}(n-k)!},$ $k=0,1,\dotsc,n$ .

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Defines:: $s_{k}(n)$ (locally)
Symbols:: : factorial (as in ), : integer and : nonnegative integer
Permalink:: http://dlmf.nist.gov/10.49.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(iv), §10.49 and Ch.10

Then

10.49.18 ${\mathsf{j}_{n}}^{2}\left(z\right)+{\mathsf{y}_{n}}^{2}\left(z\right)=\sum_{k=% 0}^{n}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}.$

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Symbols:: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, : integer, : nonnegative integer, : complex variable and $s_{k}(n)$
A&S Ref:: 10.1.27
Referenced by:: §10.49(iv)
Permalink:: http://dlmf.nist.gov/10.49.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §10.49(iv), §10.49 and Ch.10

10.49.19

${\mathsf{j}_{0}}^{2}\left(z\right)+{\mathsf{y}_{0}}^{2}\left(z\right)=z^{-2},$

${\mathsf{j}_{1}}^{2}\left(z\right)+{\mathsf{y}_{1}}^{2}\left(z\right)=z^{-2}+z% ^{-4},$

${\mathsf{j}_{2}}^{2}\left(z\right)+{\mathsf{y}_{2}}^{2}\left(z\right)=z^{-2}+3% z^{-4}+9z^{-6}.$

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Symbols:: $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind and : complex variable
A&S Ref:: 10.1.28--10.1.30
Permalink:: http://dlmf.nist.gov/10.49.E19
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §10.49(iv), §10.49 and Ch.10

10.49.20 $\left({\mathsf{i}^{(1)}_{n}}\left(z\right)\right)^{2}-\left({\mathsf{i}^{(2)}_% {n}}\left(z\right)\right)^{2}=(-1)^{n+1}\sum_{k=0}^{n}(-1)^{k}\frac{s_{k}(n+% \frac{1}{2})}{z^{2k+2}}.$

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