iconEuler Reference

Mathematical Functions

Special Functions.

This file contains special functions, like hyperbolic functions, elliptic functions, beta functions etc. It does also contain some polynomial functions.

Hyperbolic Functions

function sinh (x:numerical)

  Computes (exp(x)-exp(-x))/2
function cosh (x:numerical)

  Computes (exp(x)+exp(-x))/2
function asinh (z:numerical)

  Computes log(z+sqrt(z^2+1))
function acosh (z:numerical)

  Computes log(z+(z^2-1))
function sinc (z:numerical)

  Computes sin(x)/x
  
  Takes care of x~=0, returning 1 in this case.
  For intervals containting 0 the function fails.
function sec (x:numerical)

  Computes 1/cos(x)
function cosec (x:numerical)

  Computes 1/sin(x)
function cot (x:numerical)

  Computes 1/tan(x)

Polar Coordinates

function polar 

  polar(x,y), polar(x,y,z) compute polar coordinates.
  
  This function computes the polar coordinates in two or three
  dimensions.
  >{phi,r}=polar(1,2); degprint(phi), r,
  63°26'5.82''
  2.2360679775
  
  Returns {phi,r} or {phi,psi,r}
  
  See: 
rect (Mathematical Functions)
function rect 

  rect(phi,r), rect(phi,psi,r) compute rectangle coordinates.
  
  Computes rectangle coordinates in two or three dimensions.
  
  >{x,y}=rect(45°,sqrt(2)); [x,y]
  [ 1  1 ]
  
  Returns {x,y} or {x,y,z}
  
  See: 
polar (Mathematical Functions),
polar (Maxima Documentation)

Logarithms

function logbase (x:numerical, a:numerical)

  Computes the logarithm to base a
function log10 (x:numerical)

  Computes the logarithm to base 10

Chebychev Polynomials

function polydif (p:vector)

  Returns the polynomial p'.
  
  Polynomials are stored in Euler starting with the constant
  coefficient.
  
  >polydif([0,0,1])
  [ 0  2 ]
function cheb (x:complex, n:nonnegative integer)

  Computes the Chebyshev polynomial T_n(x).
  
  The functions used T(x,n) = cos(n*acos(x)), if possible. For x<-1
  and x>1, it uses T(x,n) = (w+1/w)/2, where w is the n-th power of
  the inverse Joukowski transform.
  
  >x=-1.1:0.01:1.1; y=x'; z=x+I*y;
  >plot2d(abs(cheb(z,5)),niveau=exp(0:10))
  
  See: 
chebpoly (Mathematical Functions),
chebrek (Mathematical Functions)
function chebrek (x:vector, n:nonnegative integer)

  Computes the Chebyshev polynomial via the recursion formula.
  
  >chebrek(-1:0.1:1,10)
  [ 1  -0.2007474688  0.9884965888  -0.0998400512  -0.9884965888  -0.5
  0.5623462912  0.9955225088  0.4284556288  -0.5388927488  -1
  -0.5388927488  0.4284556288  0.9955225088  0.5623462912  -0.5
  -0.9884965888  -0.0998400512  0.9884965888  -0.2007474688  1 ]
  
  Works for vector x and scalar n.
function chebpoly (n:nonnegative integer)

  Computes the coefficients of the n-th chebyshev polynomial.
  
  >p=chebpoly(10)
  [ -1  0  50  0  -400  0  1120  0  -1280  0  512 ]
  >plot2d("polyval(p,x)",-1,1);
  
  >x=-1.1:0.01:1.1; y=x'; z=x+I*y;
  >p=chebpoly(10); w=abs(polyval(p,z));
  >plot2d(log(w+0.2),niveau=0:10,>hue,>spectral)
  
  The function uses a loop.
function chebfit (xp:vector, yp:vector, n:nonnegative integer, ..
    a:real scalar=-1, b:real scalar=1)

  Fits Chebyshev polynomials on [a,b] to (xp,yp).
  
  Returns the vector of coefficients p(x) = sum a_[i] T(x,i)
  
  See: 
chebval (Mathematical Functions)
function chebval (x:number; alpha:vector, ..
    a:real scalar=-1, b:real scalar=1)

  Evaluates sum a[i]*T(x,i).
  
  See: 
chebfit (Mathematical Functions)

Beta, Gamma and Elliptical Functions

function beta (a:numerical, b:numerical)

  Computes the beta function
function betai (x:real, a:real, b:real)

  Computes the incomplete beta function.
  
function complexgamma (z:complex, p=gammaparams)

  Complex gamma function.
function gamma (z:numerical)

  Gamma function in the complex plane.
  
  You can also use _gamma(x) for real gamma function directly. The
  function uses an iteration implemented in the Euler language.
  
  >x=-2:0.05:2; y=x'; z=x+I*y; w=log(abs(gamma(z)));
  >plot3d(x,y,w/totalmax(w),>contour,zoom=3);
  
  See: 
gamma (Euler Core),
gamma (Maxima Documentation)
function ellrf (x:real nonnegative scalar, ..
    y:real nonnegative scalar, z:real nonnegative scalar)

  Carlson's elliptic integral of the first kind RF (x; y; z).
  
  The iteration is implemented in the Euler language.
  
  x, y, and z must be nonnegative, and at most one can be zero.
function ellf (phi: real scalar, k: real scalar)

  Elliptic integral of the first kind F(phi,k)
function ellrd (x:real nonnegative scalar, ..
    y:real nonnegative scalar, z:real nonnegative scalar)

  Carlson's elliptic integral of the second kind RD(x; y; z).
  
  x and y must be nonnegative, and at most one can be zero. z must be
  positive.
  
  See: 
elle (Mathematical Functions)
function elle (phi: real scalar, k: real scalar)

  Elliptic integral of the second kind E(phi,k).
  
  See: 
ellrd (Mathematical Functions)

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