# Special Functions

### Gamma Function

Gamma Function

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Resulting Equation

${\displaystyle ~\Gamma (z)~\Gamma (1-z)}$

${\displaystyle ~=}$

${\displaystyle ~{\frac {\pi }{\sin(\pi z)}}}$

${\displaystyle ~{\biggl |}}$

for example, if
${\displaystyle ~z\rightarrow (m-n+{\tfrac {1}{2}})}$

 ${\displaystyle ~\Rightarrow ~~~\Gamma (m-n+{\tfrac {1}{2}})~\Gamma (n-m+{\tfrac {1}{2}})}$ ${\displaystyle ~=}$ ${\displaystyle ~\pi {\biggl \{}\sin {\biggl [}{\frac {\pi }{2}}+\pi (m-n){\biggr ]}{\biggr \}}^{-1}}$ ${\displaystyle ~=}$ ${\displaystyle ~\pi (-1)^{m-n}}$
${\displaystyle ~{\biggl |}}$
Valid for:

${\displaystyle ~z\neq 0,\pm 1,\pm 2,}$

${\displaystyle ~{\biggl |}}$

### Complete Elliptic Integrals

Complete Elliptic Integral …

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… of the First Kind

 ${\displaystyle {\frac {2K(k)}{\pi }}}$ ${\displaystyle =}$ ${\displaystyle 1+{\biggl (}{\frac {1}{2}}{\biggr )}^{2}k^{2}+{\biggl (}{\frac {1\cdot 3}{2\cdot 4}}{\biggr )}^{2}k^{4}+{\biggl (}{\frac {1\cdot 3\cdot 5}{2^{4}\cdot 3}}{\biggr )}^{2}k^{6}+{\biggl (}{\frac {1\cdot 3\cdot 5\cdot 7}{2^{7}\cdot 3}}{\biggr )}^{2}k^{8}+\cdots +{\biggl [}{\frac {(2n-1)!!}{2^{n}n!}}{\biggr ]}^{2}k^{2n}+\cdots }$ Gradshteyn & Ryzhik (1965), §8.113.1

… of the First Kind (alternate expression)

 ${\displaystyle K(\mu )}$ ${\displaystyle =}$ ${\displaystyle \ln {\frac {4}{k^{'}}}+{\frac {1}{2^{2}}}{\biggl (}\ln {\frac {4}{k^{'}}}-{\frac {2}{1\cdot 2}}{\biggr )}(k')^{2}+{\biggl (}{\frac {1\cdot 3}{2\cdot 4}}{\biggr )}^{2}{\biggl (}\ln {\frac {4}{k^{'}}}-{\frac {2}{1\cdot 2}}-{\frac {2}{3\cdot 4}}{\biggr )}(k')^{4}+{\biggl (}{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}{\biggr )}^{2}{\biggl (}\ln {\frac {4}{k^{'}}}-{\frac {2}{1\cdot 2}}-{\frac {2}{3\cdot 4}}-{\frac {2}{5\cdot 6}}{\biggr )}(k')^{6}~+~\cdots }$ Gradshteyn & Ryzhik (1965), §8.113.3 where:   ${\displaystyle ~k^{'}\equiv (1-\mu ^{2})^{1/2}}$

… of the Second Kind

 ${\displaystyle {\frac {2E(k)}{\pi }}}$ ${\displaystyle =}$ ${\displaystyle 1-{\frac {1}{2^{2}}}~k^{2}-{\frac {1^{2}\cdot 3}{2^{2}\cdot 4^{2}}}~k^{4}-{\biggl (}{\frac {1\cdot 3\cdot 5}{2^{4}\cdot 3}}{\biggr )}^{2}~{\frac {k^{6}}{5}}-{\biggl (}{\frac {1\cdot 3\cdot 5\cdot 7}{2^{7}\cdot 3}}{\biggr )}^{2}{\frac {k^{8}}{7}}~-~\cdots -{\biggl [}{\frac {(2n-1)!!}{2^{n}n!}}{\biggr ]}^{2}{\frac {k^{2n}}{2n-1}}~-~\cdots }$ Gradshteyn & Ryzhik (1965), §8.114.1

… of the Second Kind (alternate expression)

 ${\displaystyle E(\mu )}$ ${\displaystyle =}$ ${\displaystyle 1~+~{\frac {1}{2}}{\biggl (}\ln {\frac {4}{k'}}-{\frac {1}{1\cdot 2}}{\biggr )}(k')^{2}~+~{\frac {1^{2}\cdot 3}{2^{2}\cdot 4}}{\biggl (}\ln {\frac {4}{k'}}-{\frac {2}{1\cdot 2}}-{\frac {1}{3\cdot 4}}{\biggr )}(k')^{4}~+~{\frac {1^{2}\cdot 3^{2}\cdot 5}{2^{2}\cdot 4^{2}\cdot 6}}{\biggl (}\ln {\frac {4}{k'}}-{\frac {2}{1\cdot 2}}-{\frac {2}{3\cdot 4}}-{\frac {1}{5\cdot 6}}{\biggr )}(k')^{6}~+~\cdots }$ Gradshteyn & Ryzhik (1965), §8.114.3 where:   ${\displaystyle k^{'}\equiv (1-\mu ^{2})^{1/2}}$

### Toroidal Function Evaluations

#### Analytic Expressions & Plots

Toroidal Function Evaluations

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Resulting Equation

 ${\displaystyle P_{-{\frac {1}{2}}}(z)}$ ${\displaystyle =}$ ${\displaystyle {\frac {2}{\pi }}{\biggl [}{\frac {2}{z+1}}{\biggr ]}^{1/2}~K{\biggl (}{\sqrt {\frac {z-1}{z+1}}}{\biggr )}}$ for example … ${\displaystyle P_{-{\frac {1}{2}}}(\cosh \eta )}$ ${\displaystyle =}$ ${\displaystyle {\biggl [}{\frac {\pi }{2}}\cdot \cosh {\frac {\eta }{2}}{\biggr ]}^{-1}K{\biggl (}\tanh {\frac {\eta }{2}}{\biggr )}}$ Abramowitz & Stegun (1995), p. 337, eq. (8.13.1) Abramowitz & Stegun (1995), p. 337, eq. (8.13.2)

NOTE: We have explicitly demonstrated that an alternate, equivalent expression is:

 ${\displaystyle P_{-{\frac {1}{2}}}(\cosh \eta )}$ ${\displaystyle =}$ ${\displaystyle {\frac {\sqrt {2}}{\pi }}(\sinh \eta )^{-1/2}kK(k)}$ where: ${\displaystyle k}$ ${\displaystyle \equiv }$ ${\displaystyle [2/(\coth \eta +1)]^{1/2}\,.}$
 ${\displaystyle Q_{-{\frac {1}{2}}}(z)}$ ${\displaystyle =}$ ${\displaystyle {\sqrt {\frac {2}{z+1}}}~K{\biggl (}{\sqrt {\frac {2}{z+1}}}{\biggr )}}$ for example … ${\displaystyle Q_{-{\frac {1}{2}}}(\cosh \eta )}$ ${\displaystyle =}$ ${\displaystyle 2e^{-\eta /2}K(e^{-\eta })}$ Abramowitz & Stegun (1995), p. 337, eq. (8.13.3) Abramowitz & Stegun (1995), p. 337, eq. (8.13.4)
 ${\displaystyle P_{+{\frac {1}{2}}}(z)}$ ${\displaystyle =}$ ${\displaystyle {\frac {2}{\pi }}{\biggl [}z+{\sqrt {z^{2}-1}}{\biggr ]}^{1/2}~E{\biggl (}{\sqrt {\frac {2(z^{2}-1)^{1/2}}{z+(z^{2}-1)^{1/2}}}}{\biggr )}}$ for example … ${\displaystyle P_{+{\frac {1}{2}}}(\cosh \eta )}$ ${\displaystyle =}$ ${\displaystyle {\frac {2}{\pi }}~e^{\eta /2}~E({\sqrt {1-e^{-2\eta }}})}$ Abramowitz & Stegun (1995), p. 337, eq. (8.13.5) Abramowitz & Stegun (1995), p. 337, eq. (8.13.6)

NOTE: It appears as though an alternate, equivalent expression is:

 ${\displaystyle P_{+{\frac {1}{2}}}(\cosh \eta )}$ ${\displaystyle =}$ ${\displaystyle {\frac {\sqrt {2}}{\pi }}(\sinh \eta )^{+1/2}k^{-1}E(k)}$ where: ${\displaystyle k}$ ${\displaystyle \equiv }$ ${\displaystyle [2/(\coth \eta +1)]^{1/2}\,.}$
 ${\displaystyle Q_{+{\frac {1}{2}}}(z)}$ ${\displaystyle =}$ ${\displaystyle z{\sqrt {\frac {2}{z+1}}}~K{\biggl (}{\sqrt {\frac {2}{z+1}}}{\biggr )}~-~[2(z+1)]^{1/2}E{\biggl (}{\sqrt {\frac {2}{z+1}}}{\biggr )}}$ Abramowitz & Stegun (1995), p. 337, eq. (8.13.7)
 ${\displaystyle Q_{-{\frac {1}{2}}}^{1}(z)}$ ${\displaystyle =}$ ${\displaystyle -{\biggl [}{\frac {1}{2(z-1)}}{\biggr ]}^{1/2}E(k)}$ where:   ${\displaystyle k={\sqrt {\frac {2}{z+1}}}\,.}$ (see our associated derivation)
 ${\displaystyle Q_{-{\frac {1}{2}}}^{2}(z)}$ ${\displaystyle =}$ ${\displaystyle {\frac {4zE(k)-(z-1)K(k)}{[2^{3}(z+1)(z-1)^{2}]^{1/2}}}}$ where:   ${\displaystyle k\equiv {\sqrt {\frac {2}{z+1}}}\,.}$ (see our associated derivation)

#### Caption for Plots

 Caption for Plots:   Here we explain how we assembled the various plots — shown immediately above in the right-hand column of the "Toroidal Function Evaluations" table — that depict the behavior of various associated Legendre (toroidal) functions (see the related discussion) having varying half-integer degrees ${\displaystyle P_{-{\frac {1}{2}}}^{0}}$, ${\displaystyle P_{+{\frac {1}{2}}}^{0}}$, ${\displaystyle Q_{-{\frac {1}{2}}}^{0}}$, ${\displaystyle Q_{+{\frac {1}{2}}}^{0}}$, ${\displaystyle Q_{+{\frac {3}{2}}}^{0}\,,}$ and (in association with a separate related discussion) having varying order ${\displaystyle Q_{-{\frac {1}{2}}}^{1}}$, ${\displaystyle Q_{-{\frac {1}{2}}}^{2}}$. For each choice of the integer indexes, ${\displaystyle n\geq 0}$ and ${\displaystyle m\geq 0}$, the relevant plot shows how the function, ${\displaystyle X_{m-{\frac {1}{2}}}^{n}(z)}$, varies with ${\displaystyle z}$. (Click on the small plot image to view an enlarged image.) In each plot … The solid green circular markers identify data that has been pulled directly from Table IX (p. 1923) of [MF53]; The solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, ${\displaystyle K(k)}$ and ${\displaystyle E(k)}$, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of [CRC]. (See an accompanying sample of elliptic integral values extracted from [CRC].) The dashed red curve was also derived using formulae expressed in terms of the complete elliptic integrals, but the values of the elliptic integrals have been calculated using (double-precision versions of) algorithms drawn from Numerical Recipes. NOTE: The tabulated values of the function, ${\displaystyle Q_{-{\frac {1}{2}}}^{1}}$, that appear in Table IX (p. 1923) of [MF53] — also see immediately below — are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this specific function — both here and in our accompanying discussion — we have plotted the absolute value of the function, ${\displaystyle |Q_{-{\frac {1}{2}}}^{1}(z)|}$. ADDITIONAL NOTE:   In Example 4 on p. 340 of Abramowitz & Stegun (1995), we can pull one additional data point for comparison; specifically, they provide a high-precision evaluation of ${\displaystyle ~Q_{-{\frac {1}{2}}}^{0}(z=2.6)=1.419337751}$. As can be seen in the table of function values immediately below, this is entirely consistent with the lower-precision value that we have extracted from [MF53], and exactly matches the double-precision value we have calculated based on the Numerical Recipes algorithm.

#### Example Recurrence Relations

The above Toroidal Function Evaluations table provides analytic expressions for the pair of foundation functions, ${\displaystyle P_{-{\frac {1}{2}}}^{0}(z)}$ and ${\displaystyle P_{+{\frac {1}{2}}}^{0}(z)}$, and the associated pair of foundation functions, ${\displaystyle Q_{-{\frac {1}{2}}}^{0}(z)}$ and ${\displaystyle Q_{+{\frac {1}{2}}}^{0}(z)}$. From either pair of foundation functions, expressions for all other zero-order, half-integer degree toroidal functions can be obtained using a relatively simple recurrence relation drawn from the "Key Equation,"

 ${\displaystyle (\nu -\mu +1)P_{\nu +1}^{\mu }(z)}$ ${\displaystyle =}$ ${\displaystyle (2\nu +1)zP_{\nu }^{\mu }(z)-(\nu +\mu )P_{\nu -1}^{\mu }(z)}$ Abramowitz & Stegun (1995), p. 334, eq. (8.5.3)
 NOTE: ${\displaystyle Q_{\nu }^{\mu }}$, as well as ${\displaystyle P_{\nu }^{\mu }}$, satisfies this same recurrence relation.

Specifically, letting ${\displaystyle \mu \rightarrow 0}$ and ${\displaystyle \nu \rightarrow (m-{\tfrac {1}{2}})}$, for all ${\displaystyle ~m\geq 2}$, we have,

 ${\displaystyle ~P0_{m-{\frac {1}{2}}}(z)}$ ${\displaystyle =}$ ${\displaystyle 4{\biggl [}{\frac {m-1}{2m-1}}{\biggr ]}zP_{m-{\frac {3}{2}}}^{0}(z)-{\biggl [}{\frac {2m-3}{2m-1}}{\biggr ]}P_{m-{\frac {5}{2}}}^{0}(z)\,;}$     and, ${\displaystyle Q_{m-{\frac {1}{2}}}^{0}(z)}$ ${\displaystyle =}$ ${\displaystyle 4{\biggl [}{\frac {m-1}{2m-1}}{\biggr ]}zQ_{m-{\frac {3}{2}}}^{0}(z)-{\biggl [}{\frac {2m-3}{2m-1}}{\biggr ]}Q_{m-{\frac {5}{2}}}^{0}(z)\,.}$

As examples, these two relations have been used to generate columns of numbers in the comparison table shown below for, respectively, the toroidal functions, ${\displaystyle P_{+{\frac {3}{2}}}^{0}(z)}$ and ${\displaystyle Q_{+{\frac {3}{2}}}^{0}(z)}$. For order-1 and order-2 toroidal functions, the above table provides analytic expressions only for (the functions of the lowest half-integer degree) ${\displaystyle Q_{-{\frac {1}{2}}}^{1}(z)}$ and ${\displaystyle Q_{-{\frac {1}{2}}}^{2}(z)}$. But, as we have detailed in an accompanying discussion, additional order-1 and order-2 expressions can be straightforwardly derived by drawing upon another key recurrence relation, namely,

 ${\displaystyle P_{\nu }^{\mu +1}(z)}$ ${\displaystyle =}$ ${\displaystyle (z^{2}-1)^{-{\frac {1}{2}}}\{(\nu -\mu )zP_{\nu }^{\mu }(z)-(\nu +\mu )P_{\nu -1}^{\mu }(z)\}}$ Abramowitz & Stegun (1995), p. 333, eq. (8.5.1)
 NOTE: ${\displaystyle Q_{\nu }^{\mu }}$, as well as ${\displaystyle P_{\nu }^{\mu }}$, satisfies this same recurrence relation.

Specifically, after adopting the association, ${\displaystyle \nu \rightarrow (n-{\tfrac {1}{2}})}$, we have, when ${\displaystyle \mu =0}$,

 ${\displaystyle Q_{n-{\frac {1}{2}}}^{1}(z)}$ ${\displaystyle =}$ ${\displaystyle (n-{\tfrac {1}{2}})(z^{2}-1)^{-{\frac {1}{2}}}[zQ_{n-{\frac {1}{2}}}(z)-Q_{n-{\frac {3}{2}}}(z)]}$ … for ${\displaystyle n\geq 1\,,}$

and, when ${\displaystyle ~\mu =1}$,

 ${\displaystyle Q_{n-{\frac {1}{2}}}^{2}(z)}$ ${\displaystyle =}$ ${\displaystyle (z^{2}-1)^{-{\frac {1}{2}}}\{(n-{\tfrac {3}{2}})zQ_{n-{\frac {1}{2}}}^{1}(z)-(n+{\tfrac {1}{2}})Q_{n-{\frac {3}{2}}}^{1}(z)\}}$ … for ${\displaystyle n\geq 1\,.}$

As an example, the first of these two relations has been used to generate a column of numbers in the comparison table shown below for the toroidal function, ${\displaystyle Q_{+{\frac {1}{2}}}^{1}(z)}$.

#### Comparison with Table IX from MF53

To facilitate copying & pasting for immediate use by other researchers, here we present in a tab-delimited, plain-text format the evaluation of nine separate toroidal functions: (Top half of table) ${\displaystyle ~P_{-{\frac {1}{2}}}^{0}}$, ${\displaystyle ~P_{+{\frac {1}{2}}}^{0}}$ and ${\displaystyle ~P_{+{\frac {3}{2}}}^{0}}$; (Bottom half of table) ${\displaystyle ~Q_{-{\frac {1}{2}}}^{0}}$, ${\displaystyle ~Q_{-{\frac {1}{2}}}^{1}}$, ${\displaystyle ~Q_{-{\frac {1}{2}}}^{2}}$, ${\displaystyle ~Q_{+{\frac {1}{2}}}^{0}}$, ${\displaystyle ~Q_{+{\frac {1}{2}}}^{1}}$ and ${\displaystyle ~Q_{+{\frac {3}{2}}}^{0}}$. Each function has been evaluated for approximately 23 different argument values in the range, ${\displaystyle ~1.0\leq z\leq 9.0}$, and, for each function, two columns of function values have been presented: (Left column) Low-precision evaluation extracted directly from Table IX (p. 1923) of [MF53]; (Right column) Our double-precision evaluation based on a set of Numerical Recipes algorithms. One exception: The value listed under the "MF53" column for the evaluation of ${\displaystyle ~Q_{-{\frac {1}{2}}}^{0}(z=2.6)}$ is the high-precision value published on p. 340 of Abramowitz & Stegun (1995); notice that our high-precision evaluation matches all ten digits of their published value.

 Top half of Table IX (p. 1923) of [MF53] z P0m1Half(z) P0p1Half(z) P0p3Half(z) MF53 Our Calc. MF53 Our Calc. MF53 Our Calc. 1.0 1.0000 1.0000 1.0000 1.2 0.9763 9.763155118E-01 1.0728 1.072784040E+00 1.3910 1.391015961E+00 1.4 0.9549 9.549467781E-01 1.1416 1.141585331E+00 1.8126 1.812643692E+00 1.6 0.9355 9.355074856E-01 1.2070 1.206963827E+00 2.2630 2.263020336E+00 1.8 0.9177 9.176991005E-01 1.2694 1.269362428E+00 2.7406 2.740570128E+00 2.0 0.9013 9.012862994E-01 1.3291 1.329138155E+00 3.2439 3.243939648E+00 2.2 0.8861 8.860804115E-01 1.3866 1.386583505E+00 3.7719 3.771951476E+00 2.4 0.8719 8.719279330E-01 1.4419 1.441941436E+00 4.3236 4.323569952E+00 2.6 0.8587 8.587023595E-01 1.4954 1.495416274E+00 4.8979 4.897875630E+00 2.8 0.8463 8.462982520E-01 1.5472 1.547181667E+00 5.4941 5.494045473E+00 3.0 0.8346 8.346268417E-01 1.5974 1.597386605E+00 6.1113 6.111337473E+00 3.5 0.8082 8.081851582E-01 1.7169 1.716877977E+00 7.7427 7.742702172E+00 4.0 0.7850 7.849616703E-01 1.8290 1.828992729E+00 9.4930 9.492973996E+00 4.5 0.7643 7.643076802E-01 1.9349 1.934919997E+00 11.3555 1.135475076E+01 5.0 0.7457 7.457491873E-01 2.0356 2.035563839E+00 13.3220 1.332184253E+01 5.5 0.7289 7.289297782E-01 2.1316 2.131629923E+00 15.3890 1.538897617E+01 6.0 0.7136 7.135750093E-01 2.2237 2.223681177E+00 17.5520 1.755159108E+01 6.5 0.6995 6.994692725E-01 2.3122 2.312174942E+00 19.8060 1.980569307E+01 7.0 0.6864 6.864402503E-01 2.3975 2.397488600E+00 22.1480 2.214774685E+01 7.5 0.6743 6.743481630E-01 2.4799 2.479937758E+00 24.5750 2.457459486E+01 8.0 0.6631 6.630781433E-01 2.5598 2.559789460E+00 27.0830 2.708339486E+01 8.5 6.525347093E-01 2.637271986E+00 2.967157094E+01 9.0 6.426376817E-01 2.712582261E+00 3.233677457E+01  Bottom half of Table IX (p. 1923) of [MF53] ATTENTION:   Widen your browser window, or "zoom out," in order to obtain a proper view of the space-delimited columns of numbers in this table. z Q0m1Half(z) Q1m1Half(z) Q2m1Half(z) Q0p1Half(z) Q1p1Half(z) Q0p3Half(z) MF53 Our Cal. MF53 Our Calc. MF53 Our Calc. M53 Our Calc. MF53 Our Calc. MF53 Our Calc. 1.1 2.8612 2.861192872E+00 2.3661 -2.366084077E+00 10.6440 1.064378304E+01 0.9788 9.787602829E-01 1.9471 -1.947110839E+00 0.4818 4.817841242E-01 1.2 2.5010 2.500956508E+00 1.7349 -1.734890983E+00 5.6518 5.651832631E+00 0.6996 6.995548314E-01 1.2524 -1.252395745E+00 0.2856 2.856355610E-01 1.4 2.1366 2.136571733E+00 1.2918 -1.291802851E+00 3.1575 3.157491205E+00 0.4598 4.597941602E-01 0.7618 -7.618218821E-01 0.14609 1.460918547E-01 1.6 1.9229 1.922920866E+00 1.0943 -1.094337965E+00 2.3230 2.323018870E+00 0.3430 3.430180260E-01 0.5501 -5.500770475E-01 0.09080 9.079816684E-02 1.8 1.7723 1.772268479E+00 0.9748 -9.748497733E-01 1.9018 1.901788930E+00 0.2720 2.720401772E-01 0.4285 -4.284853031E-01 0.06214 6.214026586E-02 2.0 1.6566 1.656638170E+00 0.8918 -8.917931374E-01 1.6454 1.645348489E+00 0.2240 2.240142929E-01 0.3489 -3.488955345E-01 0.04516 4.515872426E-02 2.2 1.5634 1.563378886E+00 0.8293 -8.292825549E-01 1.4712 1.471197798E+00 0.18932 1.893229696E-01 0.29263 -2.926294028E-01 0.03422 3.422108228E-02 2.4 1.4856 1.485653983E+00 0.7798 -7.797558474E-01 1.3441 1.344108936E+00 0.16312 1.631167365E-01 0.25076 -2.507568731E-01 0.02676 2.675556229E-02 2.6 1.419337751 1.419337751E+00 0.7391 -7.390875295E-01 1.2465 1.246521876E+00 0.14266 1.426580119E-01 0.21842 -2.184222751E-01 0.02143 2.143519083E-02 2.8 1.3617 1.361744950E+00 0.7048 -7.048053314E-01 1.1687 1.168702464E+00 0.12628 1.262756033E-01 0.19274 -1.927423405E-01 0.01751 1.751393553E-02 3.0 1.3110 1.311028777E+00 0.6753 -6.753219405E-01 1.1048 1.104816977E+00 0.11289 1.128885424E-01 0.17189 -1.718911443E-01 0.01454 1.454457729E-02 3.5 1.2064 1.206444997E+00 0.6163 -6.163068170E-01 0.9846 9.846190928E-01 0.08824 8.824567577E-02 0.13380 -1.338040913E-01 0.00966 9.664821286E-03 4.0 1.1242 1.124201960E+00 0.5713 -5.712994484E-01 0.8990 8.990205764E-01 0.07154 7.154134054E-02 0.10819 -1.081900595E-01 0.00682 6.819829619E-03 4.5 1.0572 1.057164923E+00 0.5353 -5.353494651E-01 0.8339 8.338659751E-01 0.05957 5.956966068E-02 0.08993 -8.992645608E-02 0.00503 5.029656514E-03 5.0 1.0011 1.001077380E+00 0.5057 -5.056928088E-01 0.7820 7.819717783E-01 0.05063 5.062950976E-02 0.07634 -7.633526879E-02 0.00384 3.837604899E-03 5.5 0.9532 9.532056775E-01 0.4806 -4.806378723E-01 0.7393 7.392682950E-01 0.04374 4.373774515E-02 0.06588 -6.588433822E-02 0.00301 3.008238619E-03 6.0 0.9117 9.116962715E-01 0.4591 -4.590784065E-01 0.7033 7.032568965E-01 0.03829 3.828867029E-02 0.05764 -5.763649873E-02 0.00241 2.410605139E-03 6.5 0.87524 8.752387206E-01 0.44025 -4.402537373E-01 0.67231 6.723067009E-01 0.03389 3.389003482E-02 0.05099 -5.098806037E-02 0.00197 1.967394932E-03 7.0 0.84288 8.428751774E-01 0.42362 -4.236198508E-01 0.64530 6.453008278E-01 0.03028 3.027740449E-02 0.04553 -4.553369214E-02 0.00163 1.630716095E-03 7.5 0.81389 8.138862008E-01 0.40877 -4.087751846E-01 0.62144 6.214442864E-01 0.02727 2.726650960E-02 0.04099 -4.099183107E-02 0.00137 1.369695722E-03 8.0 0.78772 7.877190099E-01 0.39542 -3.954155185E-01 0.60015 6.001530105E-01 0.02473 2.472532098E-02 0.03716 -3.716124286E-02 0.00116 1.163753807E-03 8.5 7.639406230E-01 -3.833053056E-01 5.809864341E-01 2.255696890E-02 -3.389458114E-02 9.987731857E-04 9.0 7.422062367E-01 -3.722587645E-01 5.636047532E-01 2.068890884E-02 -3.108168349E-02 8.648271474E-04 

### Relationships Between Various Associated Legendre Functions

Relationships Between Various Associated Legendre Functions

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Resulting Equation

 ${\displaystyle ~P_{\nu }^{n}(z)}$ ${\displaystyle ~=}$ ${\displaystyle ~{\frac {\Gamma (\nu +n+1)}{\Gamma (\nu -n+1)}}P_{\nu }^{-n}(z)}$ A. Erdélyi (1953):  Volume I, §3.3.1, p. 140, eq. (7)

 ${\displaystyle Q_{\nu }[tt^{'}-(t^{2}-1)^{1/2}(t^{'2}-1)^{1/2}\cos \psi ]}$ ${\displaystyle =}$ ${\displaystyle Q_{\nu }(t)P_{\nu }(t^{'})+2\sum _{n=1}^{\infty }(-1)^{n}Q_{\nu }^{n}(t)P_{\nu }^{-n}(t^{'})\cos(n\psi )}$ A. Erdélyi (1953):  Volume I, §3.11, p. 169, eq. (4)
 Valid for: ${\displaystyle t,t^{'}}$  real ${\displaystyle 1 ${\displaystyle \nu \neq -1,-2,-3,}$ … ${\displaystyle \psi }$   real
 ${\displaystyle Q_{n-1/2}^{m}(\lambda )}$ ${\displaystyle =}$ ${\displaystyle (-1)^{n}{\frac {\pi ^{3/2}}{{\sqrt {2}}~\Gamma (n-m+1/2)}}(x^{2}-1)^{1/4}P_{m-1/2}^{n}(x)\,,}$ Gil, Segura, & Temme (2000):  eq. (8)
 where: ${\displaystyle \lambda \equiv x/{\sqrt {x^{2}-1}}}$
 ${\displaystyle Q_{\nu }^{\mu }(z)}$ ${\displaystyle =}$ ${\displaystyle e^{i\mu \pi }~(2\pi )^{-{\frac {1}{2}}}(z^{2}-1)^{\mu /2}~\Gamma (\mu +{\tfrac {1}{2}})~{\biggl \{}\int _{0}^{\pi }(z-\cos t)^{-\mu -{\frac {1}{2}}}\cos[(\nu +{\tfrac {1}{2}})t]~dt-\cos(\nu \pi )\int _{0}^{\infty }(z+\cosh t)^{-\mu -{\frac {1}{2}}}e^{-(\nu +{\frac {1}{2}})t}~dt{\biggr \}}}$ A. Erdélyi (1953):  Volume I, §3.7, p. 156, eq. (10)
 Valid for: ${\displaystyle \mathrm {Re} ~\nu >-{\tfrac {1}{2}}}$ and ${\displaystyle \mathrm {Re} (\nu +\mu +1)>0\,.}$

 ${\displaystyle (\nu -\mu +1)P_{\nu +1}^{\mu }(z)}$ ${\displaystyle =}$ ${\displaystyle (2\nu +1)zP_{\nu }^{\mu }(z)-(\nu +\mu )P_{\nu -1}^{\mu }(z)}$ Abramowitz & Stegun (1995), p. 334, eq. (8.5.3)
 NOTE: ${\displaystyle Q_{\nu }^{\mu }}$, as well as ${\displaystyle P_{\nu }^{\mu }}$, satisfies this same recurrence relation.
 ${\displaystyle \int _{a}^{b}{\biggl [}(\nu -\sigma )(\nu +\sigma +1)+(\rho ^{2}-\mu ^{2})(1-z^{2})^{-1}{\biggr ]}w_{\nu }^{\mu }~w_{\sigma }^{\rho }~dz}$ ${\displaystyle =}$ ${\displaystyle {\biggl [}z(\nu -\sigma )w_{\nu }^{\mu }~w_{\sigma }^{\rho }+(\sigma +\rho )w_{\nu }^{\mu }~w_{\sigma -1}^{\rho }-(\nu +\mu )w_{\nu -1}^{\mu }~w_{\sigma }^{\rho }{\biggr ]}_{a}^{b}}$ A. Erdélyi (1953):  Volume I, §3.12, p. 169, eq. (1)
 where, ${\displaystyle w_{\nu }^{\mu }(z)}$ and ${\displaystyle w_{\sigma }^{\rho }(z)}$ denote any solutions of Legendre's differential equation
 ${\displaystyle (\xi -z)\sum _{m=0}^{n}(2m+1)P_{m}(z)Q_{m}(\xi )}$ ${\displaystyle =}$ ${\displaystyle 1-(\ell +1)[P_{\ell +1}(z)Q_{\ell }(\xi )-P_{\ell }(z)Q_{\ell +1}(\xi )]}$ Abramowitz & Stegun (1995), p. 335, eq. (8.9.2)
 ${\displaystyle P_{\nu }^{\mu +1}(z)}$ ${\displaystyle =}$ ${\displaystyle (z^{2}-1)^{-{\frac {1}{2}}}\{(\nu -\mu )zP_{\nu }^{\mu }(z)-(\nu +\mu )P_{\nu -1}^{\mu }(z)\}}$ Abramowitz & Stegun (1995), p. 333, eq. (8.5.1)
 NOTE: ${\displaystyle Q_{\nu }^{\mu }}$, as well as ${\displaystyle P_{\nu }^{\mu }}$, satisfies this same recurrence relation.
 ${\displaystyle Q_{-{\frac {1}{2}}}^{\mu }(z)+2\sum _{n=1}^{\infty }Q_{n-{\frac {1}{2}}}^{\mu }(z)\cos(n\nu )}$ ${\displaystyle =}$ ${\displaystyle e^{i\mu \pi }~{\biggl (}{\frac {\pi }{2}}{\biggr )}^{1/2}\Gamma (\mu +{\tfrac {1}{2}}){\biggl [}{\frac {(z^{2}-1)^{\mu /2}}{(z-\cos \nu )^{\mu +{\frac {1}{2}}}}}{\biggr ]}}$ A. Erdélyi (1953):  Volume I, §3.10, p. 166, eq. (3)
 Valid for: ${\displaystyle \mathrm {Re} ~\mu >-{\tfrac {1}{2}}}$

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