Compute Airy functions of the first and second kind, and their derivatives.
These routines compute the Airy function Ai(x) with an accuracy specified by mode.
These routines compute the Airy function derivative Ai'(x) with an accuracy specified by mode.
These routines compute the derivative of the scaled Airy function S_A(x) Ai(x).
These routines compute a scaled version of the Airy function S_A(x) Ai(x).
These routines compute the Airy function Bi(x) with an accuracy specified by mode.
These routines compute the Airy function derivative Bi'(x) with an accuracy specified by mode.
These routines compute the derivative of the scaled Airy function S_B(x) Bi(x).
These routines compute a scaled version of the Airy function S_B(x) Bi(x).
These routines compute the location of the s-th zero of the Airy function Ai(x).
These routines compute the location of the s-th zero of the Airy function derivative Ai(x).
These routines compute the location of the s-th zero of the Airy function Bi(x).
These routines compute the location of the s-th zero of the Airy function derivative Bi(x).
These routines compute the Arctangent integral AtanInt(x) = \\int_0^x dt \\arctan(t)/t.
Compute Bessel or Hankel functions of various kinds:
See besselj.
See besselj.
Compute Bessel or Hankel functions of various kinds:
See besselj.
See besselj.
These routines compute the scaled regular modified spherical Bessel function of order l, \\exp(-|x|) i_l(x)
These routines compute the regular modified cylindrical Bessel function of order n, I_n(x).
These routines compute the regular modified Bessel function of fractional order nu, I_\ u(x) for x>0, \ u>0.
These routines compute the scaled regular modified Bessel function of fractional order nu, \\exp(-|x|)I_\ u(x) for x>0, \ u>0.
These routines compute the scaled regular modified cylindrical Bessel function of order n, \\exp(-|x|) I_n(x)
These routines compute the regular spherical Bessel function of order l, j_l(x), for l >= 0 and x >= 0.
These routines compute the regular cylindrical Bessel function of order n, J_n(x).
These routines compute the regular cylindrical Bessel function of fractional order nu, J_\ u(x).
These routines compute the scaled irregular modified spherical Bessel function of order l, \\exp(x) k_l(x), for x>0.
These routines compute the irregular modified cylindrical Bessel function of order n, K_n(x), for x > 0.
These routines compute the irregular modified Bessel function of fractional order nu, K_\ u(x) for x>0, \ u>0.
These routines compute the scaled irregular modified Bessel function of fractional order nu, \\exp(+|x|) K_\ u(x) for x>0, \ u>0.
ERR contains an estimate of the absolute error in the value Y.
These routines compute the logarithm of the irregular modified Bessel function of fractional order nu, \\ln(K_\ u(x)) for x>0, \ u>0.
These routines compute the irregular spherical Bessel function of order l, y_l(x), for l >= 0.
These routines compute the irregular cylindrical Bessel function of order n, Y_n(x), for x>0.
These routines compute the irregular cylindrical Bessel function of fractional order nu, Y_\ u(x).
These routines compute the location of the s-th positive zero of the Bessel function J_0(x).
These routines compute the location of the s-th positive zero of the Bessel function J_1(x).
For real inputs, return the Beta function,
This function is provided for compatibility with older versions of Octave.
Return the incomplete Beta function,
Return the log of the Beta function,
These routines compute the Beta Function, B(a,b) = \\Gamma(a)\\Gamma(b)/\\Gamma(a+b) for a > 0, b > 0.
These routines compute the integral
compute the cosine integral function define by:
The Clausen function is defined by the following integral,
These routines compute the conical function P^0_{-1/2 + i \\lambda}(x) for x > -1.
These routines compute the conical function P^1_{-1/2 + i \\lambda}(x) for x > -1.
These routines compute the irregular Spherical Conical Function P^{1/2}_{-1/2 + i \\lambda}(x) for x > -1.
These routines compute the regular Spherical Conical Function P^{-1/2}_{-1/2 + i \\lambda}(x) for x > -1.
These routines compute the Wigner 3-j coefficient,
These routines compute the Wigner 6-j coefficient,
These routines compute the Wigner 9-j coefficient,
The Dawson integral is defined by \\exp(-x^2) \\int_0^x dt \\exp(t^2).
The Debye functions are defined by the integral
The Debye functions are defined by the integral
The Debye functions are defined by the integral
The Debye functions are defined by the integral
These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable mode.
These routines compute the complete elliptic integral K(k)
Compute the Jacobi elliptic functions sn(u|m), cn(u|m) and dn(u|m) for complex argument u and parameter 0 <= m <= 1.
complete elliptic integral of first K(m)
Computes the error function,
Computes the complementary error function, `1 - erf (Z)'.
compute the scaled complementary error function define by :
These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\\sqrt(\\pi)) \\int_x^\\infty \\exp(-t^2).
Computes the inverse of the error function.
These routines compute the error function erf(x) = (2/\\sqrt(\\pi)) \\int_0^x dt \\exp(-t^2).
These routines compute the upper tail of the Gaussian probability function Q(x) = (1/(2\\pi)) \\int_x^\\infty dt \\exp(-t^2/2).
These routines compute the Gaussian probability function Z(x) = (1/(2\\pi)) \\exp(-x^2/2).
These routines compute the eta function \\eta(s) for arbitrary s.
These routines compute the eta function \\eta(n) for integer n.
compute the exponential integral,
These routines compute the exponential integral Ei_3(x) = \\int_0^x dt \\exp(-t^3) for x >= 0.
compute the exponential integral,
These routines compute the second-order exponential integral E_2(x),
compute the exponential integral,
These routines compute the quantity \\exp(x)-1 using an algorithm that is accurate for small x.
These routines compute the quantity (\\exp(x)-1)/x using an algorithm that is accurate for small x.
These routines compute the quantity 2(\\exp(x)-1-x)/x^2 using an algorithm that is accurate for small x.
These routines compute the N-relative exponential, which is the n-th generalization of the functions gsl_sf_exprel and gsl_sf_exprel2.
These routines exponentiate x and multiply by the factor y to return the product y \\exp(x).
These routines compute the complete Fermi-Dirac integral F_{3/2}(x).
These routines compute the complete Fermi-Dirac integral F_{1/2}(x).
These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \\ln(1 + e^{b-x}) - (b-x).
These routines compute the complete Fermi-Dirac integral with an integer index of j, F_j(x) = (1/\\Gamma(j+1)) \\int_0^\\infty dt (t^j /(\\exp(t-x)+1)).
These routines compute the complete Fermi-Dirac integral F_{-1/2}(x).
Computes the Gamma function,
gamma(complex)
use Tim Reluga's Gamma.m or Eyal Doron's cgamma.m
This function is provided for compatibility with older versions of Octave.
Compute the normalized incomplete gamma function,
These routines compute the reciprocal of the gamma function, 1/\\Gamma(x) using the real Lanczos method.
gammaln
not implemented
These routines compute the regulated Gamma Function \\Gamma^*(x) for x > 0.
These routines compute the Gamma function \\Gamma(x), subject to x not being a negative integer.
These functions compute the incomplete Gamma Function the normalization factor included in the previously defined functions: \\Gamma(a,x) = \\int_x\\infty dt t^{a-1} \\exp(-t) for a real and x >= 0.
These routines compute the complementary normalized incomplete Gamma Function P(a,x) = 1/\\Gamma(a) \\int_0^x dt t^{a-1} \\exp(-t) for a > 0, x >= 0.
These routines compute the normalized incomplete Gamma Function Q(a,x) = 1/\\Gamma(a) \\int_x\\infty dt t^{a-1} \\exp(-t) for a > 0, x >= 0.
The hazard function for the normal distrbution, also known as the inverse Mill\\'s ratio, is defined as h(x) = Z(x)/Q(x) = \\sqrt{2/\\pi \\exp(-x^2 / 2) / \\erfc(x/\\sqrt 2)}.
These routines compute the hypergeometric function 0F1(c,x).
Primary Confluent Hypergoemetric U function A&E 13.1.3 All inputs are double as is the output.
Secondary Confluent Hypergoemetric U function A&E 13.1.3 All inputs are double as is the output.
These routines compute the Hurwitz zeta function \\zeta(s,q) for s > 1, q > 0.
Compute the Lambert W function of z.
These compute the principal branch of the Lambert W function, W_0(x).
These compute the secondary real-valued branch of the Lambert W function, W_{-1}(x).
Compute the Legendre function of degree N and order M = 0 ... N.
These functions evaluate the Legendre polynomial P_l(x) for a specific value of l, x subject to l >= 0, |x| <= 1
These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1.
These routines compute the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0.
These routines compute the normalized associated Legendre polynomial $\\sqrt{(2l+1)/(4\\pi)} \\sqrt{(l-m)!/(l+m)!} P_l^m(x)$ suitable for use in spherical harmonics.
This function computes an array of normalized associated Legendre functions sqrt((2l+1)/(4*pi)) * sqrt((l-m)!/(l+m)!) Plm (x) for m >= 0, l = |m|, ..., lmax, |x| <= 1.0
Return the natural logarithm of the gamma function of X.
These routines compute the logarithm of the Beta Function, \\log(B(a,b)) for a > 0, b > 0.
These routines compute \\log(\\cosh(x)) for any x.
These routines compute the logarithm of the Gamma function, \\log(\\Gamma(x)), subject to x not a being negative integer.
These routines compute the logarithm of the Pochhammer symbol, \\log((a)_x) = \\log(\\Gamma(a + x)/\\Gamma(a)) for a > 0, a+x > 0.
These routines compute \\log(\\sinh(x)) for x > 0.
These routines compute \\log(1 + x) for x > -1 using an algorithm that is accurate for small x.
These routines compute \\log(1 + x) - x for x > -1 using an algorithm that is accurate for small x.
These routines compute the logarithm of the complementary error function \\log(\\erfc(x)).
These routines compute the Pochhammer symbol
These routines compute the relative Pochhammer symbol ((a,x) - 1)/x where (a,x) = (a)_x := \\Gamma(a + x)/\\Gamma(a).
compute the psi function,
These routines compute the real part of the digamma function on the line 1+i y, Re[\\psi(1 + i y)].
These routines compute the Trigamma function \\psi(n) for positive integer n.
These routines compute the polygamma function \\psi^{(m)}(x) for m >= 0, x > 0.
These routines compute the integral Shi(x) = \\int_0^x dt \\sinh(t)/t.
compute the sine integral function define by:
These routines compute \\sinc(x) = \\sin(\\pi x) / (\\pi x) for any value of x.
These routines compute the first synchrotron function x \\int_x^\\infty dt K_{5/3}(t) for x >= 0.
These routines compute the second synchrotron function x K_{2/3}(x) for x >= 0.
These routines compute the Taylor coefficient x^n / n! for x >= 0, n >= 0.
These routines compute the transport function J(2,x).
These routines compute the transport function J(3,x).
These routines compute the transport function J(4,x).
These routines compute the transport function J(5,x).
compute the Riemann's Zeta function.
These routines compute the Riemann zeta function \\zeta(n) for integer n, n \ e 1.
Compute the inverse cosine in radians for each element of X.
Compute the inverse cosine in degrees for each element of X.
Compute the inverse hyperbolic cosine for each element of X.
Compute the inverse cotangent in radians for each element of X.
Compute the inverse cotangent in degrees for each element of X.
Compute the inverse hyperbolic cotangent of each element of X.
Compute the inverse cosecant in radians for each element of X.
Compute the inverse cosecant in degrees for each element of X.
Compute the inverse hyperbolic cosecant of each element of X.
Compute the inverse secant in radians for each element of X.
Compute the inverse secant in degrees for each element of X.
Compute the inverse hyperbolic secant of each element of X.
Compute the inverse sine in radians for each element of X.
Compute the inverse sine in degrees for each element of X.
Compute the inverse hyperbolic sine for each element of X.
Compute the inverse tangent in radians for each element of X.
Compute atan (Y / X) for corresponding elements of Y and X.
Compute the inverse tangent in degrees for each element of X.
Compute the inverse hyperbolic tangent for each element of X.
Compute the cosine for each element of X in radians.
Compute the cosine for each element of X in degrees.
Compute the hyperbolic cosine for each element of X.
Compute the cotangent for each element of X in radians.
Compute the cotangent for each element of X in degrees.
Compute the hyperbolic cotangent of each element of X.
Compute the cosecant for each element of X in radians.
Compute the cosecant for each element of X in degrees.
Compute the hyperbolic cosecant of each element of X.
Compute the element-by-element square root of the sum of the squares of X and Y.
Compute the secant for each element of X in radians.
Compute the secant for each element of X in degrees.
Compute the hyperbolic secant of each element of X.
Compute the sine for each element of X in radians.
Compute the sine for each element of X in degrees.
Compute the hyperbolic sine for each element of X.
Compute the tangent for each element of X in radians.
Compute the tangent for each element of X in degrees.
Compute hyperbolic tangent for each element of X.
