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

§10.49 Explicit Formulas

ⓘ

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

§10.49(i) Unmodified Functions

ⓘ

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		$a_{k}(n+\tfrac{1}{2})=\begin{cases}\dfrac{(n+k)!}{2^{k}k!(n-k)!},&k=0,1,\dotsc% ,n,\\ 0,&k=n+1,n+2,\dotsc.\end{cases}$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $n$ : integer, $k$ : 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		$\mathsf{j}_{n}\left(z\right)=\sin\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{% \left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+% \cos\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right% \rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}.$
ⓘ 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 $x$ , $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer, $k$ : nonnegative integer, $z$ : 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	$\displaystyle\mathsf{j}_{0}\left(z\right)$	$\displaystyle=\frac{\sin z}{z},$
	$\displaystyle\mathsf{j}_{1}\left(z\right)$	$\displaystyle=\frac{\sin z}{z^{2}}-\frac{\cos z}{z},$
	$\displaystyle\mathsf{j}_{2}\left(z\right)$	$\displaystyle=\left(-\frac{1}{z}+\frac{3}{z^{3}}\right)\sin z-\frac{3}{z^{2}}% \cos z.$
ⓘ 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 $z$ : 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		$\mathsf{y}_{n}\left(z\right)=-\cos\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{% \left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+1}}+% \sin\left(z-\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right% \rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+2}}.$
ⓘ 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 $x$ , $\sin\NVar{z}$ : sine function, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $n$ : integer, $k$ : nonnegative integer, $z$ : 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	$\displaystyle\mathsf{y}_{0}\left(z\right)$	$\displaystyle=-\frac{\cos z}{z},$
	$\displaystyle\mathsf{y}_{1}\left(z\right)$	$\displaystyle=-\frac{\cos z}{z^{2}}-\frac{\sin z}{z},$
	$\displaystyle\mathsf{y}_{2}\left(z\right)$	$\displaystyle=\left(\frac{1}{z}-\frac{3}{z^{3}}\right)\cos z-\frac{3}{z^{2}}% \sin z.$
ⓘ 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 $z$ : 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	$\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)$	$\displaystyle=e^{iz}\sum_{k=0}^{n}i^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}},$
ⓘ 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, $n$ : integer, $k$ : nonnegative integer, $z$ : 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	$\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)$	$\displaystyle=e^{-iz}\sum_{k=0}^{n}(-i)^{k-n-1}\frac{a_{k}(n+\frac{1}{2})}{z^{% k+1}}.$
ⓘ 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, $n$ : integer, $k$ : nonnegative integer, $z$ : 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

ⓘ

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		${\mathsf{i}^{(1)}_{n}}\left(z\right)=\tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}% \frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n+1}\*\tfrac{1}{2}e^{-z}\sum_{k=0}^% {n}\frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}.$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $k$ : nonnegative integer, $z$ : 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	$\displaystyle{\mathsf{i}^{(1)}_{0}}\left(z\right)$	$\displaystyle=\frac{\sinh z}{z},$
	$\displaystyle{\mathsf{i}^{(1)}_{1}}\left(z\right)$	$\displaystyle=-\frac{\sinh z}{z^{2}}+\frac{\cosh z}{z},$
	$\displaystyle{\mathsf{i}^{(1)}_{2}}\left(z\right)$	$\displaystyle=\left(\frac{1}{z}+\frac{3}{z^{3}}\right)\sinh z-\frac{3}{z^{2}}% \cosh z.$
ⓘ 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 $z$ : 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		${\mathsf{i}^{(2)}_{n}}\left(z\right)=\tfrac{1}{2}e^{z}\sum_{k=0}^{n}(-1)^{k}% \frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}+(-1)^{n}\tfrac{1}{2}e^{-z}\sum_{k=0}^{n}% \frac{a_{k}(n+\frac{1}{2})}{z^{k+1}}.$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $k$ : nonnegative integer, $z$ : 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	$\displaystyle{\mathsf{i}^{(2)}_{0}}\left(z\right)$	$\displaystyle=\frac{\cosh z}{z},$
	$\displaystyle{\mathsf{i}^{(2)}_{1}}\left(z\right)$	$\displaystyle=-\frac{\cosh z}{z^{2}}+\frac{\sinh z}{z},$
	$\displaystyle{\mathsf{i}^{(2)}_{2}}\left(z\right)$	$\displaystyle=\left(\frac{1}{z}+\frac{3}{z^{3}}\right)\cosh z-\frac{3}{z^{2}}% \sinh z.$
ⓘ 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 $z$ : 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		$\mathsf{k}_{n}\left(z\right)=\tfrac{1}{2}\pi e^{-z}\sum_{k=0}^{n}\frac{a_{k}(n% +\frac{1}{2})}{z^{k+1}}.$
ⓘ 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, $n$ : integer, $k$ : nonnegative integer, $z$ : 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	$\displaystyle\mathsf{k}_{0}\left(z\right)$	$\displaystyle=\tfrac{1}{2}\pi\frac{e^{-z}}{z},$
	$\displaystyle\mathsf{k}_{1}\left(z\right)$	$\displaystyle=\tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{1}{z^{2}}\right),$
	$\displaystyle\mathsf{k}_{2}\left(z\right)$	$\displaystyle=\tfrac{1}{2}\pi e^{-z}\left(\frac{1}{z}+\frac{3}{z^{2}}+\frac{3}% {z^{3}}\right).$
ⓘ 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 $z$ : 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 $n$ . 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

ⓘ

Keywords:: Rayleigh’s formulas, spherical Bessel functions
Notes:: For the first of(10.49.14) combine the second of (10.51.3), with $n=0$ and $m=n$ , 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	$\displaystyle\mathsf{j}_{n}\left(z\right)$	$\displaystyle=z^{n}\left(-\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}% \frac{\sin z}{z},$
10.49.14	$\displaystyle\mathsf{y}_{n}\left(z\right)$	$\displaystyle=-z^{n}\left(-\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n% }\frac{\cos z}{z}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\sin\NVar{z}$ : 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, $n$ : integer and $z$ : 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	$\displaystyle{\mathsf{i}^{(1)}_{n}}\left(z\right)$	$\displaystyle=z^{n}\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}% \frac{\sinh z}{z},$
10.49.15	$\displaystyle{\mathsf{i}^{(2)}_{n}}\left(z\right)$	$\displaystyle=z^{n}\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}% \frac{\cosh z}{z}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\cosh\NVar{z}$ : 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, $n$ : integer and $z$ : 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		$\mathsf{k}_{n}\left(z\right)=(-1)^{n}\tfrac{1}{2}\pi z^{n}\left(\frac{1}{z}% \frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}\frac{e^{-z}}{z}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : 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

ⓘ

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		$s_{k}(n+\tfrac{1}{2})=\frac{(2k)!(n+k)!}{2^{2k}(k!)^{2}(n-k)!},$
$k=0,1,\dotsc,n$ .
ⓘ Defines: $s_{k}(n)$ (locally) Symbols: $!$ : factorial (as in $n!$ ), $n$ : integer and $k$ : 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		${\mathsf{j}_{n}}^{2}\left(z\right)+{\mathsf{y}_{n}}^{2}\left(z\right)=\sum_{k=% 0}^{n}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}.$
ⓘ 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, $n$ : integer, $k$ : nonnegative integer, $z$ : 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	$\displaystyle{\mathsf{j}_{0}}^{2}\left(z\right)+{\mathsf{y}_{0}}^{2}\left(z\right)$	$\displaystyle=z^{-2},$
	$\displaystyle{\mathsf{j}_{1}}^{2}\left(z\right)+{\mathsf{y}_{1}}^{2}\left(z\right)$	$\displaystyle=z^{-2}+z^{-4},$
	$\displaystyle{\mathsf{j}_{2}}^{2}\left(z\right)+{\mathsf{y}_{2}}^{2}\left(z\right)$	$\displaystyle=z^{-2}+3z^{-4}+9z^{-6}.$
ⓘ 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 $z$ : 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		$\left({\mathsf{i}^{(1)}_{n}}\left(z\right)\right)^{2}-\left({\mathsf{i}^{(2)}_% {n}}\left(z\right)\right)^{2}=(-1)^{n+1}\sum_{k=0}^{n}(-1)^{k}\frac{s_{k}(n+% \frac{1}{2})}{z^{2k+2}}.$
ⓘ Symbols: ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer, $k$ : nonnegative integer, $z$ : complex variable and $s_{k}(n)$ A&S Ref: 10.2.26 Referenced by: §10.49(iv) Permalink: http://dlmf.nist.gov/10.49.E20 Encodings: TeX, pMML, png See also: Annotations for §10.49(iv), §10.49 and Ch.10

10.49.21	$\displaystyle\Big{(}{\mathsf{i}^{(1)}_{0}}\left(z\right)\Big{)}^{2}-\Big{(}{% \mathsf{i}^{(2)}_{0}}\left(z\right)\Big{)}^{2}$	$\displaystyle=-z^{-2},$
	$\displaystyle\Big{(}{\mathsf{i}^{(1)}_{1}}\left(z\right)\Big{)}^{2}-\Big{(}{% \mathsf{i}^{(2)}_{1}}\left(z\right)\Big{)}^{2}$	$\displaystyle=z^{-2}-z^{-4},$
	$\displaystyle\Big{(}{\mathsf{i}^{(1)}_{2}}\left(z\right)\Big{)}^{2}-\Big{(}{% \mathsf{i}^{(2)}_{2}}\left(z\right)\Big{)}^{2}$	$\displaystyle=-z^{-2}+3z^{-4}-9z^{-6}.$
ⓘ Symbols: ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function, ${\mathsf{i}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : modified spherical Bessel function and $z$ : complex variable A&S Ref: 10.2.27--10.2.29 Permalink: http://dlmf.nist.gov/10.49.E21 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §10.49(iv), §10.49 and Ch.10

10.48 Graphs 10.50 Wronskians and Cross-Products

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

§10.49 Explicit Formulas

Contents

§10.49(i) Unmodified Functions

§10.49(ii) Modified Functions

§10.49(iii) Rayleigh’s Formulas

§10.49(iv) Sums or Differences of Squares

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.