Legendre polynomials and related#

Legendre polynomial (or function) of the first kind, \(P_n(x)\)#

ctx.legendre_p(n, x)#

where ctx is math53, ctxboost or ctxflint.

Returns \(\displaystyle P_n(x) = \,_2F_1\left(-n, n+1, 1, \frac{1-x}{2}\right)\), the Legendre polynomial of degree \(n\). The Legendre polynomials are orthogonal on the interval \((-1, 1)\) with \(w(x) = 1\). If \(n \geq 0\) the function uses the following recurrence relation, with \(P_n(x) = P_{-n-1}(x)\):

\begin{eqnarray} P_0 (x) & = & 1 \\ P_1 (x) & = & x \nonumber \\ (n+1)P_{n+1} (x)& = & (2n+1) P_{n}(x) - n P_{n-1}(x). \nonumber \end{eqnarray}

See also Wikipedia [1420], MathWorld [1002], NIST [305], BoostMath [131], Ehrhardt [309] (3.7.13), Flint [825], Flint [821], Mpmath [666].

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.LegendreP(3, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.LegendreP(6, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.LegendreP(3, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.LegendreP(6, '0.51')
Gpr('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n = '10'; x = '5.0'
>>> \mathrm{d}x = dec.legendre(n, x); mx = mpm.legendre(n, x); gx = gmp.legendre(n, x)
>>> fx = fpm.legendre(n, x); ax = apm.legendre(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  1.600472677000000000000000000000000000000E+9
mpm:  1.600472677000000000000000000000000000000e+9
gmp:  1.600472677000000000000000000000000000000E+09
fpm:  1.60047267700000E+09
apm:  1.600472677000000000000000000000000000000e+9 (1.54e-39%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n = '10'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.legendre(n, z); mz = mpm.legendre(n, z); gz = gmp.legendre(n, z)
>>> fz = fpm.legendre(n, z); az = apm.legendre(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 5.4324745448476562500E+9        - 5.7861538517578125000E+9j
mpm: 5.4324745448476562500e+9        - 5.7861538517578125000e+9j
gmp: 5.4324745448476562500E+09       - 5.7861538517578125000E+09j
fpm: 5.43247454484766E+09            - 5.78615385175781E+09j
apm: 5.4324745448476562500e+9 (0.0%) - 5.7861538517578125000e+9 (0.0%)j

Associated Legendre function of the first kind, \(P^m_l(x)\)#

ctx.legendre_plm(l, m, x)#

where ctx is math53, ctxboost or ctxflint.

Returns \(\displaystyle \frac{1}{\Gamma(1-m)} \frac{(1+z)^{m/2}}{(1-z)^{m/2}} \,_2F_1\left(-n, n+1, 1-m, \frac{1-z}{2}\right).\), the (associated) Legendre function of the first kind of degree \(n\) and order \(m\). Taking \(m = 0\) gives the ordinary Legendre function of the first kind, \(P_n(z)\).

See also Wikipedia [1401], MathWorld [1001], NIST [305], BoostMath [131], Ehrhardt [309] (3.7.14), Flint [825], Flint [821], Mpmath [654].

Many different branch cut conventions appear in the literature. If type is 0, the version

\[P_n^m(z) = \frac{(1+z)^{m/2}}{(1-z)^{m/2}} \mathbf{F}\left(-n, n+1, 1-m, \frac{1-z}{2}\right)\]

is computed, and if type is 1, the alternative version

\[{\mathcal P}_n^m(z) = \frac{(z+1)^{m/2}}{(z-1)^{m/2}} \mathbf{F}\left(-n, n+1, 1-m, \frac{1-z}{2}\right).\]

is computed. Type 0 and type 1 respectively correspond to type 2 and type 3 in Mathematica and mpmath.

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.LegendrePlm(2, 3, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.LegendrePlm(2, 6, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.LegendrePlm(2, 3, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.LegendrePlm(2, 6, '0.51')
Gpr('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n = '10'; m = '7'; x = '0.5'
>>> \mathrm{d}x = dec.legenp(n, m, x); mx = mpm.legenp(n, m, x); gx = gmp.legenp(n, m, x)
>>> fx = fpm.legenp(n, m, x); ax = apm.legenp(n, m, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  -1.836027792321961447325321009149754756328E+6
mpm:  -1.836027792321961447325321009149754756328e+6
gmp:  -1.836027792321961447325321009149754756328E+06
fpm:  -1.83602779232196E+06
apm:  -1.836027792321961447325321009149754756328e+6 (-6.556e-40%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n = '10 + 0j'; m = '7 + 1j'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.legenp(n, m, z); mz = mpm.legenp(n, m, z); gz = gmp.legenp(n, m, z)
>>> fz = fpm.legenp(n, m, z); az = apm.legenp(n, m, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 5.1676243722757488933E+14              - 9.9095186680298524778E+14j
mpm: 5.1676243722757488933e+14              - 9.9095186680298524778e+14j
gmp: 5.1676243722757488933E+14              - 9.9095186680298524778E+14j
fpm: 5.16762437227575E+14                   - 9.90951866802985E+14j
apm: 5.1676243722757488933e+14 (2.768e-19%) - 9.9095186680298524778e+14 (-1.925e-19%)j

Legendre function of the second kind, \(Q_l(x)\)#

ctx.legendre_q(l, x)#

where ctx is math53, ctxboost or ctxflint.

Returns \(\displaystyle Q_n(x) = \,_2F_1\left(\frac{l+1}{2}, \frac{l+2}{2}; l+\frac{3}{2}; \frac{1}{x^2}\right)\), the Legendre function of the second kind of degree \(l\). For integer \(l \geq 0\) and \(x \ne 1\), the following recurrence relations hold:

\begin{eqnarray} Q_0 (x) & = & \frac{1}{2} \log \left(\frac{1+x}{1-x}\right) \\ Q_1 (x) & = & \frac{x}{2} \log \left(\frac{1+x}{1-x}\right) -1 \nonumber \\ (k+1)Q_{k+1} (x)& = & (2k+1) Q_{k}(x) - k Q_{k-1}(x). \nonumber \end{eqnarray}

See also Wikipedia [1420], MathWorld [1002], NIST [305], BoostMath [131], Ehrhardt [309] (3.7.15), Flint [825], Flint [821], Mpmath [655].

This function returns \(Q^m_l (x)\), the associated Legendre functions of the second kind with \(l \geq 0\), \(l+m \geq 0\) and \(x \neq 1\), defined as

\[Q^m_l (x) = (-1)^m (1-x^2)^{m/2} \frac{d^m}{\mathrm{d}x^m} Q_{l} (x), \quad |x|<1,\]

\[Q^m_l (x) = (x^2-1)^{m/2} \frac{d^m}{\mathrm{d}x^m} Q_{l} (x), \quad |x|>1.\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.LegendreQ(3, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.LegendreQ(6, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.LegendreQ(3, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.LegendreQ(6, '0.51')
Gpr('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n = '10'; m = '7'; x = '0.5'
>>> \mathrm{d}x = dec.legenq(n, m, x); mx = mpm.legenq(n, m, x); gx = gmp.legenq(n, m, x)
>>> fx = fpm.legenq(n, m, x); ax = apm.legenq(n, m, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  2.716651409660037234094986783336728254923E+6
mpm:  2.716651409660037234094986783336728254923e+6
gmp:  2.716651409660037234094986783336728254923E+06
fpm:  2.71665140966004E+06
apm:  2.716651409660037234094986783336728254923e+6 (3.811e-38%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n = '10 + 0j'; m = '7 + 1j'; z = '5.0 + 3j'
>>> \mathrm{d}z = dec.legenq(n, m, z); mz = mpm.legenq(n, m, z); gz = gmp.legenq(n, m, z)
>>> fz = fpm.legenq(n, m, z); az = apm.legenq(n, m, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 1.5565835524046748818E+15              + 8.1172853822265297587E+14j
mpm: 1.5565835524046748818e+15              + 8.1172853822265297587e+14j
gmp: 1.5565835524046748818E+15              + 8.1172853822265297587E+14j
fpm: 1.55658355240468E+15                   + 8.11728538222653E+14j
apm: 1.5565835524046748818e+15 (9.619e-18%) + 8.1172853822265297586e+14 (1.78e-17%)j

Associated Legendre function of the second kind, \(Q^m_l(x)\)#

ctx.legendre_qlm(l, m, x)#

where ctx is math53, ctxboost or ctxflint.

For generalization to hypergeometric functions, see https://en.wikipedia.org/wiki/Legendre_function#Solutions_of_the_differential_equation

and https://en.wikipedia.org/wiki/Associated_Legendre_polynomials#Generalization_via_hypergeometric_functions

Returns \(\displaystyle Q_l^m(z) = \frac{\pi}{2 \sin(\pi m)} \left( \cos(\pi m) P_l^m(z) - \frac{\Gamma(1+m+l)}{\Gamma(1-m+l)} P_l^{-m}(z)\right)\), the (associated) Legendre function of the second kind of degree \(l\) and order \(m\).

Here \(\displaystyle P_l^m(z)\) is the Legendre function of the first kind of degree \(l\) and order \(m\). The formula above should be understood as limit when \(m\) is an integer. Taking \(m = 0\) gives the ordinary Legendre function of the second kind, \(Q_n(z)\).

A different formula from Wikipedia:

\[Q_{\lambda}^{\mu}(z) = \frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}\Gamma(\lambda+3/2)}\frac{1}{z^{\lambda+\mu+1}}(1-z^2)^{\mu/2} \,_2F_1 \left(\frac{\lambda+\mu+1}{2}, \frac{\lambda+\mu+2}{2}; \lambda+\frac{3}{2}; \frac{1}{z^2}\right)\]

For integer \(m\), \(l \geq 0\), \(l+m \geq 0\) and \(x \neq 1\), the function can be defined by

\[Q^m_l (x) = (-1)^m (1-x^2)^{m/2} \frac{d^m}{\mathrm{d}x^m} Q_{l} (x),\]

\[Q^{-m}_l (x) = (x^2-1)^{m/2} \frac{(l-m)!}{(l+m)!} Q_{l}^m (x).\]

The factor \((-1)^m\) is omitted if \(|x|>1\), see NIST [305] (14.9.14).

See also Wikipedia [1420], MathWorld [1002], NIST [305], BoostMath [131], Ehrhardt [309] (3.7.16), Mpmath [655].

Sets res to the associated Legendre function of the second kind evaluated for degree n, order m, and argument z. When m is zero, this reduces to the Legendre function \(Q_n(z)\).

Many different branch cut conventions appear in the literature. If type is 0, the version

\[Q_n^m(z) = \frac{\pi}{2 \sin(\pi m)} \left( \cos(\pi m) P_n^m(z) - \frac{\Gamma(1+m+n)}{\Gamma(1-m+n)} P_n^{-m}(z)\right)\]

is computed, and if type is 1, the alternative version

\[\mathcal{Q}_n^m(z) = \frac{\pi}{2 \sin(\pi m)} e^{\pi i m} \left( \mathcal{P}_n^m(z) - \frac{\Gamma(1+m+n)}{\Gamma(1-m+n)} \mathcal{P}_n^{-m}(z)\right)\]

is computed. Type 0 and type 1 respectively correspond to type 2 and type 3 in Mathematica and mpmath.

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.LegendrePlm(2, 3, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.LegendrePlm(2, 6, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.LegendrePlm(2, 3, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.LegendrePlm(2, 6, '0.51')
Gpr('5.3518479027559984754E-1')

Spherical harmonics, \(Y_n^m(\theta, \phi)\)#

ctx.spherical_harmonic(theta, phi, n, m)#

where ctx is math53, ctxboost or ctxflint.

Note: math53.spherHarm(theta, phi, n, m), ctxboost.SphericalHarmonicR(theta, phi, n, m), ctxboost.SphericalHarmonicI(theta, phi, n, m)

!!! n and m need to be integer !!!

Returns \(\displaystyle Y_l^m(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos \theta) e^{i m \phi}\) the spherical harmonic, where \(\displaystyle P_l^m(z)\) is the Legendre function of the first kind of degree \(l\) and order \(m\), \(\theta \in [0, \pi]\) denotes the polar coordinate (ranging from the north pole to the south pole) and \(\phi \in [0, 2 \pi]\) denotes the azimuthal coordinate on a sphere. Care should be used since many different conventions for spherical coordinate variables are used.

See also Wikipedia [1423], MathWorld [1008], NIST [306], BoostMath [136], Ehrhardt [309] (3.7.17), Flint [821], Mpmath [667].

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.SpherHarm(2, 3, 5, 4)
xreal('5.2359877559829887307E-1')
>>> xreal.SpherHarm(2.1, 3.1, 5.1, 4.1)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.SpherHarm(2, 3, 5, 4)
Gpr('5.2359877559829887307E-1')
>>> Gpr.SpherHarm(2.1, 3.1, 5.1, 4.1)
Gpr('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; l = '10'; m = '7'; theta = '0.1'; phi = '0.2'
>>> \mathrm{d}x = dec.spherharm(l, m, theta, phi); mx = mpm.spherharm(l, m, theta, phi);
>>> gx = gmp.spherharm(l, m, theta, phi)
>>> fx = fpm.spherharm(l, m, theta, phi); ax = apm.spherharm(l, m, theta, phi)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax], aligned=True)
dec: -2.5484347403024134998E-7               - 1.4775528280770230150E-6j
mpm: -2.5484347403024134998e-7               - 1.4775528280770230150e-6j
gmp: -2.5484347403024134998E-07              - 1.4775528280770230150E-06j
fpm: -2.54843474030241E-07                   - 1.47755282807702E-06j
apm: -2.5484347403024134998e-7 (-9.509e-19%) - 1.4775528280770230150e-6 (-4.92e-19%)j

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n = '10'; m = '7'; theta = '0.1 + 03j'; phi = '0.2 + 04j'
>>> \mathrm{d}z = dec.spherharm(l, m, theta, phi); mz = mpm.spherharm(l, m, theta, phi);
>>> gz = gmp.spherharm(l, m, theta, phi)
>>> fz = fpm.spherharm(l, m, theta, phi); az = apm.spherharm(l, m, theta, phi)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: -5.0629719914753183582E-2               + 1.2051935898383241506E-1j
mpm: -5.0629719914753183582e-2               + 1.2051935898383241506e-1j
gmp: -5.0629719914753183582E-02              + 1.2051935898383241506E-01j
fpm: -5.06297199147532E-02                   + 1.20519358983832E-01j
apm: -5.0629719914753183582e-2 (-6.797e-19%) + 1.2051935898383241506e-1 (3.075e-19%)j

Toroidal harmonics \(P^m_{l-1/2}(x)\)#

math53.toroidal_plm(l, m, x)#

Returns the toroidal harmonic \(P^m_{l-1/2}(x)\), which is an associated Legendre function \(P^m_l(x)\) of the first kind with half-integer degree. The current implementation is based on Amath and is restricted to \(l,m=0,1; x>1\), using Legendre elliptic integrals or Bulirsch elliptic integrals for numerical evaluation.

Toroidal harmonics \(Q^m_{l-1/2}(x)\)#

math53.toroidal_qlm(l, m, x)#

Returns the toroidal harmonic \(Q^m_{l-1/2}(x)\), which is an associated Legendre function \(Q^m_l(x)\) of the second kind with half-integer degree. The current implementation is based on Amath and is restricted to \(l,m=0,1; x>1\), using Legendre elliptic integrals or Bulirsch elliptic integrals for numerical evaluation.

See also: Majic (2019), https://mathworld.wolfram.com/ToroidalFunction.html

Olver’s associated Legendre function \(Q^m_{l-1/2}(x)\)#

math53.olver_qlm(l, m, x)#

Defined as (see https://dlmf.nist.gov/14.3#E10)

\[\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i}\frac{Q^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}.\]

Can be calculated as

\[\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{2^{\nu}\Gamma\left(\nu+1\right)(x+1)^{\mu/2}}{(x-1)^{(\mu/2)+\nu+1}}\mathbf{F}\left(\nu+1,\nu+\mu+1;2\nu+2;\frac{2}{1-x}\right).\]

For hypergeometric representations of Ferrers function and associated Legendre function

see https://dlmf.nist.gov/14.3

Legendre polynomials and related

Contents

Legendre polynomials and related#

Legendre polynomial (or function) of the first kind, \(P_n(x)\)#

Associated Legendre function of the first kind, \(P^m_l(x)\)#

Legendre function of the second kind, \(Q_l(x)\)#

Associated Legendre function of the second kind, \(Q^m_l(x)\)#

Spherical harmonics, \(Y_n^m(\theta, \phi)\)#

Toroidal harmonics \(P^m_{l-1/2}(x)\)#

Toroidal harmonics \(Q^m_{l-1/2}(x)\)#

Olver’s associated Legendre function \(Q^m_{l-1/2}(x)\)#