Maths Special Function

Calculate special function values.

Elliptic integrals are calculated using Carlsons forms. Jacobi elliptic functions are calculated using Landens transformation. The Gamma function is calculated using the Lanczos approximation.

Reference : Numerical Recipes, The Art Of Scientific Computing, Press, Teukolsky, Vetterling, Flannery, Cambridge University Press

Change Module :

Maths Beta Function Calculation Module
Maths Elliptic Function Calculation Module
Maths Elliptic Integral Calculation Module
Maths Error Function Calculation Module
Maths Gamma Function Calculation Module
Maths Probability Function Calculation Module
All

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Links : ±

CALCULATOR : Complete Elliptic Integral [FREE] ±

Calculate the complete Legrendre elliptic integrals of the first, second and third kind (K, E and P) from the elliptic k modulus.

Elliptic integrals are calculated for an ellipse of the form

`x^2 + (y / b)^2 = 1 `
`k = √(1 - 1 / b^2) `

where :

k = the elliptic k modulus

For a circle k = 0. k tends to 1 as b tends to infinity. The complete integrals are invalid for n greater than or equal to 1, or as k tends to 1. Use the Result Plot option to plot the complete integrals versus the k modulus.

Tool Input

k : Elliptical k Modulus
n : Elliptical n Modulus

Tool Output

E : Complete Elliptical Integral Second Kind
K : Complete Elliptical Integral First Kind
P : Complete Elliptical Integral Third Kind

CALCULATOR : Gamma Function (Lanczos Approximation) [FREE] ±

Calculate the Gamma function Γ(z), the log Gamma function ln(Γ(z)), the incomplete lower Gamma function γ(z,x), the incomplete upper Gamma function Γ(z,x), the incomplete lower Gamma unit function PL(z,x), the incomplete upper Gamma unit function PU(z,x).

The Gamma function is invalid if z equals zero, or if z is a negative integer. Use the Result Plot option to plot the Gamma functions versus z.

Tool Input

z : Z Input
x : X Input

Tool Output

Γ(z) : Gamma Function
Γ(z,x) : Upper Incomplete Gamma Function
γ(z,x) : Lower Incomplete Gamma Function
PL(z,x) : Lower Incomplete Unit Gamma Function (γ(z,x) / Γ(z))
PU(z,x) : Upper Incomplete Unit Gamma Function (Γ(z,x) / Γ(z))
chk : Incomplete Gamma Function Check Value (= 1)
ln(Γ(z)) : Log Gamma Function

CALCULATOR : Jacobi Elliptic Functions And Elliptic Amplitude [FREE] ±

Calculate the Jacobi elliptic functions, sn, cn, dn and am.

The elliptic functions are invalid for n greater than or equal to 1, or as k tends to 1. Use the Result Plot option to plot the elliptic functions versus the k modulus.

Tool Input

k : Elliptical k Modulus
u : Elliptical Arc Length

Tool Output

X : Elliptical X Coordinate
Y : Elliptical Y Coordinate
am : Elliptical Amplitude
b : Elliptical b Coeffiicent
chk : Elliptical Check Value (= 1)
cn : Elliptical cn
dn : Elliptical dn
rn : Elliptical Radius
sn : Elliptical sn

CALCULATOR : Incomplete Elliptic Integral [FREE] ±

Calculate the incomplete Legrendre elliptic integrals of the first, second and third kind (F, E and Π).

The incomplete integrals are invalid for n greater than or equal to 1, or as k tends to 1. Use the Result Plot option to plot the incomplete integrals versus the k modulus, or the ellipse part angle (≤ π / 2).

Tool Input

a : Ellipse Part Angle (≤ π / 2)
k : Elliptical k Modulus
n : Elliptical n Modulus

Tool Output

E : Incomplete Elliptical Integral Second Kind
F : Incomplete Elliptical Integral First Kind
P : Incomplete Elliptical Integral Third Kind

CALCULATOR : Error Function [FREE] ±

Calculate the error function (erf) and complementary error function (erfc).

The error function asymptotes to 1 as x tends to infinity. The complementary error function asymptotes to 0 as x tends to infinity.

Tool Input

x : Input x Value

Tool Output

erf(x) : Error Function
erfc(x) : Complementary Error Function

CALCULATOR : Normal Probability Distribution [FREE] ±

Calculate the normal probability density, cumulative distribution function and complementary distribution function from the mean, standard deviation and percentile.

The cumulative probability can be calculated from the percentile, or the percentile can be calculated from the cumulative probability. The cumulative distribution (cdf) function asymptotes to 1 as the percentile tends to infinity. The complementary distribution function asymptotes to 0 as the percentile x tends to infinity. Use the Result Plot option to plot the probability density, cumulative distribution, and complementary distribution (or tail) versus the percentile.

Tool Input

pxtype : Probability Type
- xu : User Defined Percentile
- pu : User Defined Cumulative Probability
m : Mean Value
s : Standard Deviation

Tool Output

d : Probability Density
p : Cumulative Probability
q : Tail Probability Or Exceedance
x : Percentile

CALCULATOR : Log Normal Probability Distribution [FREE] ±

Calculate the log normal probability density, cumulative distribution function and complementary distribution function from the mean, standard deviation and percentile.

The cumulative probability can be calculated from the percentile, or the percentile can be calculated from the cumulative probability. The arithmetic mean, arithmetic standard deviation, median and mode are also calculated. The cumulative distribution (cdf) function asymptotes to 1 as the percentile tends to infinity. The complementary distribution function asymptotes to 0 as the percentile x tends to infinity. Use the Result Plot option to plot the probability density, cumulative distribution, and complementary distribution (or tail) versus the percentile.

Tool Input

pxtype : Probability Type
- xu : User Defined Percentile
- pu : User Defined Cumulative Probability
m : Mean Value
s : Standard Deviation

Tool Output

am : Arithmetic Mean
as : Arithmetic Standard Deviation
d : Probability Density
med : Median
mod : Mode
p : Cumulative Probability
q : Tail Probability Or Exceedance
x : Percentile

CALCULATOR : Beta Function [FREE] ±

Calculate the Beta function B(z, w) versus z and w.

The Beta function tends to infinity for z equals zero, or if z is a negative integer.

Tool Input

z : Z Input
w : W Input

Tool Output

B(z,w) : Beta Function

CALCULATOR : Lower Incomplete Gamma Unit Function [FREE] ±

Calculate the lower incomplete Gamma unit function from z and x, or calculate the inverse function x from z and p.

The inverse function calculates x such that P(z,x) = p. The inverse function is valid for 0 ≤ p ≤ 1.

Tool Input

gztype : Gamma Function Type
- xu : User Defined Gamma x Value
- pu : User Defined Incomplete Gamma Unit Value
z : Gamma z Value

Tool Output

p : Incomplete Gamma Unit Value
x : Gamma x Value

CALCULATOR : Upper Incomplete Gamma Unit Function [FREE] ±

Calculate the upper incomplete Gamma unit function from z and x, or calculate the inverse function x from z and p.

The inverse function calculates x such that P(z,x) = p. The inverse function is valid for 0 ≤ p ≤ 1.

Tool Input

gztype : Gamma Function Type
- xu : User Defined Gamma x Value
- pu : User Defined Incomplete Gamma Unit Value
z : Gamma z Value

Tool Output

p : Incomplete Gamma Unit Value
x : Gamma x Value

CALCULATOR : Inverse Error Function [FREE] ±

Calculate the inverse error function (inverse erf) and inverse complementary error function (inverse erfc).

The inverse error function and the inverse complementary error function are valid for the range 0 ≤ e ≤ 1.

Tool Input

e : Input e Value

Tool Output

inv(e) : Inverse Error Function
invc(e) : Inverse Complementary Error Function