Additional elliptic integrals

Additional elliptic integrals#

Legendre complete elliptic integral \(B(k)\)#

math53.elliptic_b(k)#

Returns the Legendre complete elliptic integral \(\displaystyle B(k) = \int_0^{\pi/2} \frac{ \cos^2(t) \mathrm{d}t}{\sqrt{1-k^2 \sin^2(t)}} = \frac{E(k)-\sqrt{1-k^2} K(k)}{k^2}\). See also Wikipedia [1406], MathWorld [1025], NIST [178], BoostMath [143], Ehrhardt [309] (3.2.1.4).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.CompEllintB(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.CompEllintB(0.5)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.CompEllintB(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.CompEllintB(0.5)
Gpr('5.3518479027559984754E-1')

Legendre complete elliptic integral \(D(k)\)#

math53.elliptic_d(m)#

Returns the Legendre complete elliptic integral \(\displaystyle D(k) = \int_0^{\pi/2} \frac{ \sin^2(t) \mathrm{d}t}{\sqrt{1-k^2 \sin^2(t)}} = \frac{K(k) - E(k)}{k^2}\). See also Wikipedia [1406], MathWorld [1025], NIST [178], BoostMath [143], Ehrhardt [309] (3.2.1.5).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.CompEllintD(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.CompEllintD(0.5)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.CompEllintD(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.CompEllintD(0.5)
Gpr('5.3518479027559984754E-1')

Legendre incomplete elliptic integral \(B(\phi, k)\)#

math53.elliptic_b_inc(phi, m)#

Returns the Legendre complete elliptic integral , \(\displaystyle B(k) = \int_0^{\phi} \frac{ \cos^2(t) \mathrm{d}t}{\sqrt{1-k^2 \sin^2(t)}} = \frac{E(\phi, k)-\sqrt{1-k^2} F(\phi, k)}{k^2}\). See also Wikipedia [1406], MathWorld [1025], NIST [178], BoostMath [143], Ehrhardt [309] (3.2.1.9).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllintB(0.12, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllintB(0.12, 0.5)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllintB(0.12, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllintB(0.12, 0.5)
Gpr('5.3518479027559984754E-1')

Legendre incomplete elliptic integral \(D(\phi, k)\)#

math53.elliptic_d_inc(phi, m)#

Returns the Legendre complete elliptic integral , \(\displaystyle D(k) = \int_0^{\phi} \frac{ \sin^2(t) \mathrm{d}t}{\sqrt{1-k^2 \sin^2(t)}} = \frac{F(\phi, k)-E(\phi, k)}{k^2}\). See also Wikipedia [1406], MathWorld [1025], NIST [178], BoostMath [140], Ehrhardt [309] (3.2.1.10).

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.EllintD(0.12, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.EllintD(0.12, 0.5)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.EllintD(0.12, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.EllintD(0.12, 0.5)
Gpr('5.3518479027559984754E-1')

Heuman’s Lambda function, \(\Lambda_0(\phi,k)\)#

math53.heuman_lambda(phi, k)#

Returns Heuman’s Lambda function \(\Lambda_0(\phi,k)\). See also MathWorld [1032], BoostMath [147], Ehrhardt [309] (3.2.6).

\[\Lambda_0(\phi,k) = \frac{F(\phi, \sqrt{1-k^2})}{K(\sqrt{1-k^2})} + \frac{2}{\pi} K(k) Z(\phi, \sqrt{1-k^2}) , \quad |k| \le 1.\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.HeumanLambda(0.8, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.HeumanLambda(0.8, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.HeumanLambda(0.8, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.HeumanLambda(0.8, '0.51')
Gpr('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; phi = '0.3'; m = '0.7'
>>> \mathrm{d}x = dec.heuman_lambda(phi, m); mx = mpm.heuman_lambda(phi, m); gx = gmp.heuman_lambda(phi, m)
>>> fx = fpm.heuman_lambda(phi, m); ax = apm.heuman_lambda(phi, m)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  2.346706205795989147266165089161797155226E-1
mpm:  2.346706205795989147266165089161797155226e-1
gmp:  2.346706205795989147266165089161797155226E-01
fpm:  2.34670620579599E-01
apm:  2.346706205795989147266165089161797156135e-1 (4.07e-34%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; phi = '7.0 + 3.0j'; m = '11.0 + 3.0j'
>>> \mathrm{d}z = dec.heuman_lambda(phi, m); mz = mpm.heuman_lambda(phi, m); gz = gmp.heuman_lambda(phi, m)
>>> fz = fpm.heuman_lambda(phi, m); az = apm.heuman_lambda(phi, m)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: -1.1345920286482140753E+1               - 6.3944316931048323564E+0j
mpm: -1.1345920286482140753e+1               - 6.3944316931048323564e+0j
gmp: -1.1345920286482140753E+01              - 6.3944316931048323564E+00j
fpm: -1.13459202864821E+01                   - 6.39443169310483E+00j
apm: -1.1345920286482157538e+1 (-5.813e-13%) - 6.3944316931048395085e+0 (-5.749e-13%)j

Jacobi Zeta function, \(Z(\phi,k)\)#

math53.jacobi_zeta(phi, k)#

Returns the Jacobi Zeta function \(Z(\phi,k)\). See also MathWorld [1040], BoostMath [158], Ehrhardt [309] (3.2.7).

\[Z(\phi,k) = E(\phi,k) - \frac{E(k)}{K(k)} F(\phi,k), \quad |k| \le 1.\]

An example in Python

>>> from xlcalcnet import xreal
>>> xreal.JacobiZeta(0.8, 0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.JacobiZeta(0.8, '0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.JacobiZeta(0.8, 0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.JacobiZeta(0.8, '0.51')
Gpr('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; phi = '0.3'; m = '0.7'
>>> \mathrm{d}x = dec.jacobi_zeta(phi, m); mx = mpm.jacobi_zeta(phi, m); gx = gmp.jacobi_zeta(phi, m)
>>> fx = fpm.jacobi_zeta(phi, m); ax = apm.jacobi_zeta(phi, m)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  1.154857111268923923475571636423888535609E-1
mpm:  1.154857111268923923475571636423888535609e-1
gmp:  1.154857111268923923475571636423888535609E-01
fpm:  1.15485711126892E-01
apm:  1.154857111268923923475571636423888535611e-1 (3.902e-37%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; phi = '7.0 + 3.0j'; m = '11.0 + 3.0j'
>>> \mathrm{d}z = dec.jacobi_zeta(phi, m); mz = mpm.jacobi_zeta(phi, m); gz = gmp.jacobi_zeta(phi, m)
>>> fz = fpm.jacobi_zeta(phi, m); az = apm.jacobi_zeta(phi, m)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 1.7328120796787527800E+1              + 2.7272085822071339357E+1j
mpm: 1.7328120796787527800e+1              + 2.7272085822071339357e+1j
gmp: 1.7328120796787527800E+01             + 2.7272085822071339357E+01j
fpm: 1.73281207967875E+01                  + 2.72720858220713E+01j
apm: 1.7328120796787527800e+1 (1.063e-16%) + 2.7272085822071339356e+1 (8.617e-17%)j
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