13 Confluent Hypergeometric Functions Whittaker Functions 13.14 Definitions and Basic Properties 13.16 Integral Representations

§13.15 Recurrence Relations and Derivatives

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Permalink:: http://dlmf.nist.gov/13.15
See also:: Annotations for Ch.13

§13.15(i) Recurrence Relations
§13.15(ii) Differentiation Formulas

§13.15(i) Recurrence Relations

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Keywords:: Whittaker functions, recurrence relations
Notes:: See Slater (1960, §2.5). Note that (2.5.4) and (2.5.10) contain errors: the correct versions are (13.15.2) and (13.15.10), respectively.
Referenced by:: §13.29(iv)
Permalink:: http://dlmf.nist.gov/13.15.i
See also:: Annotations for §13.15 and Ch.13

13.15.1 $(\kappa-\mu-\tfrac{1}{2})M_{\kappa-1,\mu}\left(z\right)+(z-2\kappa)M_{\kappa,% \mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})M_{\kappa+1,\mu}\left(z\right)=0,$

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Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
A&S Ref:: 13.4.29
Permalink:: http://dlmf.nist.gov/13.15.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.2 $2\mu(1+2\mu)\sqrt{z}M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-(z+2% \mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}M_{% \kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,$

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Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
Referenced by:: §13.15(i)
Permalink:: http://dlmf.nist.gov/13.15.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.3 $(\kappa-\mu-\tfrac{1}{2})M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)+% (1+2\mu)\sqrt{z}M_{\kappa,\mu}\left(z\right)-(\kappa+\mu+\tfrac{1}{2})M_{% \kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,$

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Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.4 $2\mu M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-2\mu M_{\kappa+\frac% {1}{2},\mu-\frac{1}{2}}\left(z\right)-\sqrt{z}M_{\kappa,\mu}\left(z\right)=0,$

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Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
A&S Ref:: 13.4.28
Permalink:: http://dlmf.nist.gov/13.15.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.5 $2\mu(1+2\mu)M_{\kappa,\mu}\left(z\right)-2\mu(1+2\mu)\sqrt{z}M_{\kappa-\frac{1% }{2},\mu-\frac{1}{2}}\left(z\right)-(\kappa-\mu-\tfrac{1}{2})\sqrt{z}M_{\kappa% -\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,$

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Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.6 $2\mu(1+2\mu)\sqrt{z}M_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)+(z-2% \mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa-\mu-\tfrac{1}{2})\sqrt{z}M_{% \kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,$

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Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.7 $2\mu(1+2\mu)\sqrt{z}M_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-2\mu(% 1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}M_{\kappa% +\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0.$

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Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.8 $W_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)-\sqrt{z}W_{\kappa,\mu}% \left(z\right)+(\kappa-\mu-\tfrac{1}{2})W_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}% \left(z\right)=0,$

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Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.9 $W_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-\sqrt{z}W_{\kappa,\mu}% \left(z\right)+(\kappa+\mu-\tfrac{1}{2})W_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}% \left(z\right)=0,$

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Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
A&S Ref:: 13.4.30 (in different form)
Permalink:: http://dlmf.nist.gov/13.15.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.10 $2\mu W_{\kappa,\mu}\left(z\right)-\sqrt{z}W_{\kappa+\frac{1}{2},\mu+\frac{1}{2% }}\left(z\right)+\sqrt{z}W_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)=0,$

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Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
Referenced by:: §13.15(i)
Permalink:: http://dlmf.nist.gov/13.15.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.11 $W_{\kappa+1,\mu}\left(z\right)+(2\kappa-z)W_{\kappa,\mu}\left(z\right)+(\kappa% -\mu-\tfrac{1}{2})(\kappa+\mu-\tfrac{1}{2})W_{\kappa-1,\mu}\left(z\right)=0,$

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Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
A&S Ref:: 13.4.31
Permalink:: http://dlmf.nist.gov/13.15.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.12 $(\kappa-\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z% \right)+2\mu W_{\kappa,\mu}\left(z\right)-(\kappa+\mu-\tfrac{1}{2})\sqrt{z}W_{% \kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)=0,$

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Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.13 $(\kappa+\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z% \right)-(z+2\mu)W_{\kappa,\mu}\left(z\right)+\sqrt{z}W_{\kappa+\frac{1}{2},\mu% +\frac{1}{2}}\left(z\right)=0,$

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Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

13.15.14 $(\kappa-\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z% \right)-(z-2\mu)W_{\kappa,\mu}\left(z\right)+\sqrt{z}W_{\kappa+\frac{1}{2},\mu% -\frac{1}{2}}\left(z\right)=0.$

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Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(i), §13.15 and Ch.13

§13.15(ii) Differentiation Formulas

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Keywords:: Whittaker functions, derivatives
Notes:: See Slater (1960, §§2.4, 2.4.1).
Permalink:: http://dlmf.nist.gov/13.15.ii
See also:: Annotations for §13.15 and Ch.13

13.15.15 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z^{\mu-\frac{1% }{2}}M_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}{\left(-2\mu\right)_{n}}e^{% \frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}M_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}% \left(z\right),$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.16 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z^{-\mu-\frac{% 1}{2}}M_{\kappa,\mu}\left(z\right)\right)=\frac{{\left(\frac{1}{2}+\mu-\kappa% \right)_{n}}}{{\left(1+2\mu\right)_{n}}}e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+% 1)}M_{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z\right),$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.17 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{\frac{1}{2}z}z^{-% \kappa-1}M_{\kappa,\mu}\left(z\right)\right)={\left(\tfrac{1}{2}+\mu-\kappa% \right)_{n}}e^{\frac{1}{2}z}z^{n-\kappa-1}M_{\kappa-n,\mu}\left(z\right),$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.18 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}z^{\mu-\frac{% 1}{2}}M_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}{\left(-2\mu\right)_{n}}e^{-% \frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}M_{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}% \left(z\right),$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.19 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}z^{-\mu-\frac% {1}{2}}M_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}\frac{{\left(\frac{1}{2}+% \mu+\kappa\right)_{n}}}{{\left(1+2\mu\right)_{n}}}e^{-\frac{1}{2}z}z^{-\mu-% \frac{1}{2}(n+1)}\*M_{\kappa+\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z\right),$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.20 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{-\frac{1}{2}z}z^{% \kappa-1}M_{\kappa,\mu}\left(z\right)\right)={\left(\tfrac{1}{2}+\mu+\kappa% \right)_{n}}e^{-\frac{1}{2}z}z^{\kappa+n-1}\*M_{\kappa+n,\mu}\left(z\right).$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.21 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z^{-\mu-\frac{% 1}{2}}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}{\left(\tfrac{1}{2}+\mu-% \kappa\right)_{n}}e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}\*W_{\kappa-\frac{1% }{2}n,\mu+\frac{1}{2}n}\left(z\right),$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.22 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z^{\mu-\frac{1% }{2}}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}{\left(\tfrac{1}{2}-\mu-% \kappa\right)_{n}}e^{\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}\*W_{\kappa-\frac{1}% {2}n,\mu-\frac{1}{2}n}\left(z\right),$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.23 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{\frac{1}{2}z}z^{-% \kappa-1}W_{\kappa,\mu}\left(z\right)\right)={\left(\tfrac{1}{2}+\mu-\kappa% \right)_{n}}{\left(\tfrac{1}{2}-\mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{n-% \kappa-1}W_{\kappa-n,\mu}\left(z\right),$

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.24 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}z^{-\mu-\frac% {1}{2}}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}e^{-\frac{1}{2}z}z^{-\mu-% \frac{1}{2}(n+1)}W_{\kappa+\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z\right),$

ⓘ

Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.25 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}z^{\mu-\frac{% 1}{2}}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}e^{-\frac{1}{2}z}z^{\mu-% \frac{1}{2}(n+1)}W_{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z\right),$

ⓘ

Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E25
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

13.15.26 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{-\frac{1}{2}z}z^{% \kappa-1}W_{\kappa,\mu}\left(z\right)\right)=(-1)^{n}e^{-\frac{1}{2}z}z^{% \kappa+n-1}W_{\kappa+n,\mu}\left(z\right).$

ⓘ

Symbols:: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §13.15(ii), §13.15 and Ch.13

Other versions of several of the identities in this subsection can be constructed by use of (13.3.29).

13.14 Definitions and Basic Properties 13.16 Integral Representations

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§13.15 Recurrence Relations and Derivatives

Contents

§13.15(i) Recurrence Relations

§13.15(ii) Differentiation Formulas