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Maths Special Function Modules

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CALCULATOR MODULE : 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

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CALCULATOR MODULE : Maths Elliptic Integral   ±

Calculate the complete and incomplete elliptic integrals of the first, second and third kind 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. Use the Result Plot option to plot the integrals versus the k modulus.

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

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CALCULATOR MODULE : Maths Elliptic Function   ±

Calculate the elliptic functions and elliptic amplitude (sn, cn, dn and am).

Elliptic functions 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. Use the Result Plot option to plot the elliptic functions versus the k modulus.

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

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CALCULATOR MODULE : Maths Gamma Function   ±

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 a continuous form of the integer factorial:

`n! = Γ(n + 1) `

The Gamma function is recursive for values greater than 1 or less than 0 (z > 1 or z < 0).

`Γ(z + 1) = z Γ(z) `

The Gamma function is invalid if z equals zero, or if z is a negative integer. The lower incomplete Gamma unit functions are defined as

`PL(z,x) = γ(z,x) / Γ(z) `
`PU(z,x) = Γ(z,x) / Γ(z) `

Use the Result Plot option to plot the Gamma functions versus z.

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

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CALCULATOR MODULE : Maths Error Function   ±

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.

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

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CALCULATOR MODULE : Maths Probability Function   ±

Calculate normal and 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 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.

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

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CALCULATOR MODULE : Maths Beta Function   ±

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

The Beta function is calculated from the Gamma function

`B(z, w) = (Γ(z) x Γ(w)) / Γ(z + w) `

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

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

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