Any time a function allows a complex number z it also allows a real number x.
Airy Functions
- (Ai,Aip,Bi,Bip) = airy(z)
- airy(z) calculates the Airy functions and their derivatives
evaluated at real or complex number z. The Airy functions Ai and Bi
are two independent solutions of y''(x)=xy. Aip and Bip are the first derivatives
evaluated at x of Ai and Bi respectively.
- (Aie,Aipe,Bie,Bipe) = airye(z)
- airye(z) calculates the exponentially scaled Airy functions and
their derivatives evaluated at real or complex number z.
airye(z)[0:1] = airy(z)[0:1] * exp(2.0/3.0*z*sqrt(z))
airye(z)[2:3] = airy(z)[2:3] * exp(-abs((2.0/3.0*z*sqrt(z)).real))
(Apt,Bpt,Ant,Bnt) = itairy(x)
itairy(x) calculates the integral of Airy functions from 0 to x
for positive (Apt, Bpt) and negative (Ant, Bnt) arguments.
Elliptic Functions and Integrals
- (sn,cn,dn,ph) = ellipj(u,m)
- ellipj(u,m) calculates the Jacobian elliptic functions of
parameter m between 0 and 1, and real u. The returned functions are
often written sn(u|m), cn(u|m), and dn(u|m). The value of ph is such
that if u = ellik(ph,m), then sn(u|m) = sin(ph) and cn(u|m) = cos(ph).
- y = ellipe(m)
- ellipe(m) returns the complete integral of the second kind:
integral(sqrt(1-m*sin(t)**2),t=0..pi/2)
- y = ellipeinc(phi,m)
- ellipeinc(phi,m) returns the incomplete elliptic integral of the
second kind: integral(sqrt(1-m*sin(t)**2),t=0..phi)
- y = ellipk(m)
- ellipk(m) returns the complete integral of the first kind:
integral(1/sqrt(1-m*sin(t)**2),t=0..pi/2)
- y = ellipkinc(phi,m)
- ellipkinc(phi,m) returns the incomplete elliptic integral of the first
kind: integral(1/sqrt(1-m*sin(t)**2),t=0..phi)
Bessel Functions
- y = jn(n,x)
- jn(n,x) returns the Bessel function of integer order n at x.
- y = jv(v,z)
- jv(v,z) returns the Bessel function of real order v at complex z.
- y = jve(v,z)
- jve(v,z) returns the exponentially scaled Bessel function of real order
v at complex z: jve(v,z) = jv(v,z) * exp(-abs(z.imag))
- y = yn(n,x)
- yn(n,x) returns the Bessel function of the second kind of integer
order n at x.
- y = yv(n,z)
- yv(v,z) returns the Bessel function of the second kind of real
order v at complex z.
- y = yve(n,z)
- yve(v,z) returns the exponentially scaled Bessel function of the second
kind of real order v at complex z: yve(v,z) = yv(v,z) * exp(-abs(z.imag))
- y = kn(n,x)
- kn(n,x) returns the modified Bessel function of the third kind for
integer order n at x.
- y = kv(v,z)
- kv(v,z) returns the modified Bessel function of the third kind for
real order v at complex z.
- y = kve(v,z)
- kve(v,z) returns the exponentially scaled, modified Bessel function
of the third kind for real order v at complex z: kve(v,z) = kv(v,z) * exp(z)
- y = iv(v,z)
- iv(v,z) returns the modified Bessel function of real order v of
z. If z is of real type and negative, v must be integer valued.
- y = ive(v,z)
- ive(v,z) returns the exponentially scaled modified Bessel function of
real order v and complex z: ive(v,z) = iv(v,z) * exp(-abs(z.real))
- y = hankel1(v,z)
- hankel1(v,z) returns the Hankel function of the first kind for real order v and complex argument z.
- y = hankel1e(v,z)
- hankel1e(v,z) returns the exponentially scaled Hankel function of the first
kind for real order v and complex argument z:
hankel1e(v,z) = hankel1(v,z) * exp(-1j * z)
- y = hankel2(v,z)
- hankel2(v,z) returns the Hankel function of the second kind for real order v and complex argument z.
- y = hankel2e(v,z)
- hankel2e(v,z) returns the exponentially scaled Hankel function of the second
kind for real order v and complex argument z:
hankel1e(v,z) = hankel1(v,z) * exp(1j * z)
- y = j0(x)
- j0(x) returns the Bessel function of order 0 at x.
- y = j1(x)
- j1(x) returns the Bessel function of order 1 at x.
- y = y0(x)
- y0(x) returns the Bessel function of the second kind of order 0 at x.
- y = y1(x)
- y1(x) returns the Bessel function of the second kind of order 1 at x.
- y = i0(x)
- i0(x) returns the modified Bessel function of order 0 at x.
- y = i1(x)
- i1(x) returns the modified Bessel function of order 1 at x.
- y = i0e(x)
- i0e(x) returns the exponentially scaled modified Bessel function
of order 0 at x. i0e(x) = exp(-|x|) * i0(x).
- y = i1e(x)
- i1e(x) returns the exponentially scaled modified Bessel function
of order 0 at x. i1e(x) = exp(-|x|) * i1(x).
- y = k0(x)
- i0(x) returns the modified Bessel function of the third kind of
order 0 at x.
- y = k1(x)
- i1(x) returns the modified Bessel function of the third kind of
order 1 at x.
- y = k0e(x)
- k0e(x) returns the exponentially scaled modified Bessel function
of the third kind of order 0 at x. k0e(x) = exp(x) * k0(x).
- y = k1e(x)
- k1e(x) returns the exponentially scaled modified Bessel function
of the third kind of order 1 at x. k1e(x) = exp(x) * k1(x)
y = besselpoly(a,lambda,nu)
besselpoly(a,lambda,nu) computes the integral of a bessel function
times a power: y = integral(x^lambda jv(2*a*x),x=0..1)
(ij0,iy0) = itj0y0(x)
itj0y0(x) returns simple integrals from 0 to x of the zeroth order
bessel functions j0 and y0.
(ij0,iy0) = it2j0y0(x)
it2j0y0(x) returns the integrals int((1-j0(t))/t,t=0..x) and
int(y0(t)/t,t=x..infinitity).
(ii0,ik0) = iti0k0(x)
iti0k0(x) returns simple integrals from 0 to x of the zeroth order
modified bessel functions i0 and k0.
(ii0,ik0) = it2i0k0(x)
it2i0k0(x) returns the integrals int((i0(t)-1)/t,t=0..x) and
int(k0(t)/t,t=x..infinitity).
Statistical Functions
- y = bdtr(k,n,p)
- bdtr(k,n,p) returns the sum of the terms 0 through k of the
Binomial probability density: sum(nCj p**j (1-p)**(n-j),j=0..k)
- y = bdtrc(k,n,p)
- bdtrc(k,n,p) returns the sum of the terms k+1 through n of the
Binomial probability density: sum(nCj p**j (1-p)**(n-j), j=k+1..n)
- p = bdtri(k,n,y)
- bdtri(k,n,y) finds the probability p such that the sum of the
terms 0 through k of the Binomial probability density is equal to the
given cumulative probability y.
- y = btdtr(a,b,x)
- btdtr(a,b,x) returns the area from zero to x under the beta
density function: gamma(a+b)/(gamma(a)*gamma(b)))*integral(t**(a-1)
(1-t)**(b-1), t=0..x). SEE ALSO betainc
- x = btdtri(a,b,p)
- btdtri(a,b,p) returns the pth quantile of the beta distribution. It is
effectively the inverse of btdtr returning the value of x for which
btdtr(a,b,x) = p. SEE ALSO betaincinv
- y = fdtr(dfn,dfd,x)
- fdtr(dfn,dfd,x) returns the area from zero to x under the F density
function (also known as Snedcor's density or the variance ratio
density). This is the density of X = (unum/dfn)/(uden/dfd), where unum and
uden are random variables having Chi square distributions with dfn and
dfd degrees of freedom, respectively.
- y = fdtrc(dfn,dfd,x)
- fdtrc(dfn,dfd,x) returns the complemented F distribution function.
- x = fdtri(dfn,dfd,p)
- fdtri(dfn,dfd,p) finds the F density argument x such that
fdtr(dfn,dfd,x)=p.
- x = fdtridfn(dfn,dfd,p)
- fdtridfn(p,dfd,x) finds the F density argument dfn such that
fdtr(dfn,dfd,x)=p.
- x = fdtridfd(dfn,p,x)
- fdtridfd(dfn,p,x) finds the F density argument dfd such that
fdtr(dfn,dfd,x)=p.
- y = gdtr(a,b,x)
- gdtr(a,b,x) returns the integral from zero to x of the gamma
probability density function: a**b / gamma(b) * integral(t**(b-1) exp(-at),t=0..x).
The arguments a and b are used differently here than in other definitions.
- y = gdtrc(a,b,x)
- gdtrc(a,b,x) returns the integral from x to infinity of the gamma
probability density function. SEE gdtr, gdtri
- x = gdtri(a,b,p)
- gdtri(a,b,p) returns pth quantile of the gamma distribution. It is
the inverse of the gamma cdf returning the value of x for which
gdtr(b,a,x) = p.
- y = nbdtr(k,n,p)
- nbdtr(k,n,p) returns the sum of the terms 0 through k of the
negative binomial distribution: sum((n+j-1)Cj p**n (1-p)**j,j=0..k).
In a sequence of Bernoulli trials this is the probability that k or
fewer failures precede the nth success.
- y = nbdtrc(k,n,p)
- nbdtrc(k,n,p) returns the sum of the terms k+1 to infinity of the
negative binomial distribution.
- p = nbdtri(k,n,y)
- nbdtri(k,n,y) finds the argument p such that nbdtr(k,n,p)=y.
- k = nbdtrik(y,n,p)
- nbdtrik(y,n,p) finds the argument k such that nbdtr(k,n,p)=y.
- n = nbdtrin(k,y,p)
- nbdtrin(k,y,p) finds the argument n such that nbdtr(k,n,p)=y.
- y = pdtr(k,m)
- pdtr(k,m) returns the sum of the first k terms of the Poisson
distribution: sum(exp(-m) * m**j / j!, j=0..k) = gammaincc( k+1, m).
Arguments must both be positive and k an integer.
- y = pdtrc(k,m)
- pdtrc(k,m) returns the sum of the terms from k+1 to infinity of the
Poisson distribution: sum(exp(-m) * m**j / j!, j=k+1..inf) = gammainc( k+1, m).
Arguments must both be positive and k an integer.
- m = pdtri(k,y)
- pdtri(k,y) returns the Poisson variable m such that the sum
from 0 to k of the Poisson density is equal to the given probability
y: calculated by gammaincinv( k+1, y). k must be a nonnegative integer and
y between 0 and 1.
- k = pdtrik(p,m)
- pdtrik(p,m) returns the quantile k such that pdtr(k,m)=p
- p = stdtr(df,t)
- stdtr(df,t) returns the integral from minus infinity to t of the Student t
distribution with df > 0 degrees of freedom:
gamma((df+1)/2)/(sqrt(df*pi)*gamma(df/2)) * integral((1+x**2/df)**(-df/2-1/2),
x=-inf..t)
- t = stdtrit(df,p)
- stdtrit(df,p) returns the argument t such that stdtr(df,t) is equal to p.
- t = stdtridf(p,t)
- stdtridf(p,t) returns the argument df such that stdtr(df,t) is equal to p.
- p = chdtr(v,x)
- chdtr(v,x) Returns the area under the left hand tail (from 0 to x) of the Chi
square probability density function with v degrees of freedom:
1/(2**(v/2) * gamma(v/2)) * integral(t**(v/2-1) * exp(-t/2), t=0..x)
- p = chdtrc(v,x)
- chdtrc(v,x) returns the area under the right hand tail (from x to
infinity) of the Chi square probability density function with v
degrees of freedom:
1/(2**(v/2) * gamma(v/2)) * integral(t**(v/2-1) * exp(-t/2), t=x..inf)
- x = chdtri(v,p)
- chdtri(v,p) returns the argument x such that chdtrc(v,x) is equal
to p.
- y = ndtr(x)
- ndtr(x) returns the area under the standard Gaussian probability
density function, integrated from minus infinity to x:
1/sqrt(2*pi) * integral(exp(-t**2 / 2),t=-inf..x)
- x = ndtri(y)
- ndtri(y) returns the argument x for which the area udnder the
Gaussian probability density function (integrated from minus infinity
to x) is equal to y.
- y = smirnov(n,e)
- smirnov(n,e) returns the exact Kolmogorov-Smirnov complementary
cumulative distribution function (Dn+ or Dn-) for a one-sided test of
equality between an empirical and a theoretical distribution. It is equal
to the probability that the maximum difference between a theoretical
distribution and an empirical one based on n samples is greater than e.
- e = smirnovi(n,y)
- smirnovi(n,y) returns e such that smirnov(n,e) = y.
- p = kolmogorov(y)
- kolmogorov(y) returns the complementary cumulative distribution
function of Kolmogorov's limiting distribution (Kn* for large n)
of a two-sided test for equality between an empirical and a theoretical
distribution. It is equal to the (limit as n->infinity of the) probability
that sqrt(n) * max absolute deviation > y.
- y = kolmogi(p)
- kolmogi(p) returns y such that kolmogorov(y) = p
Gamma and Related Functions
- y = gamma(z)
- gamma(z) returns the gamma function of the argument. The gamma
function is often referred to as the generalized factorial since
z*gamma(z) = gamma(z+1) and gamma(n+1) = n! for natural number n.
- y = gammaln(z)
- gammaln(z) returns the base e logarithm of the absolute value of the
gamma function of z: ln(|gamma(z)|)
- y = gammainc(a,x)
- gammainc(a,x) returns the incomplete gamma integral defined as
1 / gamma(a) * integral(exp(-t) * t**(a-1), t=0..x). Both arguments
must be positive.
- y = gammaincc(a,x)
- gammaincc(a,x) returns the complemented incomplete gamma integral
defined as 1 / gamma(a) * integral(exp(-t) * t**(a-1), t=x..inf) = 1 -
gammainc(a,x). Both arguments must be positive.
- x = gammainccinv(a,y)
- gammainccinv(a,y) returns x such that gammaincc(a,x) = y.
- y = beta(a,b)
- beta(a,b) returns gamma(a) * gamma(b) / gamma(a+b)
- y = betaln(x)
- betaln(a,b) returns the natural logarithm of the absolute value of
beta: ln(|beta(x)|).
- y = betainc(a,b,x)
- betainc(a,b,x) returns the incomplete beta integral of the
arguments, evaluated from zero to x: gamma(a+b) / (gamma(a)*gamma(b))
* integral(t**(a-1) (1-t)**(b-1), t=0..x).
- x = betaincinv(a,b,y)
- betaincinv(a,b,y) returns x such that betainc(a,b,x) = y.
- y = psi(z)
- psi(z) is the derivative of the logarithm of the gamma function
evaluated at z (also called the digamma function).
- y = rgamma(z)
- rgamma(z) returns one divided by the gamma function of x.
Error Function and Fresnel integrals
- y = erf(z)
- erf(z) returns the error function of complex argument defined as
as 2/sqrt(pi)*integral(exp(-t**2),t=0..z)
- y = erfc(x)
- erfc(x) returns 1 - erf(x).
- (ssa,cca) = fresnel(z)
- fresnel(z) returns the fresnel sin and cos integrals: integral(sin(pi/2
* t**2),t=0..z) and integral(cos(pi/2 * t**2),t=0..z) for real or
complex z.
- (fp,kp) = modfresnelp(x)
- modfresnelp(x) returns the modified fresnel integrals F_+(x) and K_+(x)
as fp=integral(exp(1j*t*t),t=x..inf) and kp=1/sqrt(pi)*exp(-1j*(x*x+pi/4))*fp
- (fm,km) = modfresnelm(x)
- modfresnelp(x) returns the modified fresnel integrals F_-(x) amd K_-(x)
as fp=integral(exp(-1j*t*t),t=x..inf) and kp=1/sqrt(pi)*exp(1j*(x*x+pi/4))*fp
HyperGeometric Functions
- y = hyp2f1(a,b,c,z)
- hyp2f1(a,b,c,z) returns the gauss hypergeometric function
( 2F1(a,b;c;z) ).
- y = hyp1f1(a,b,x)
- hyp1f1(a,b,x) returns the confluent hypergeometeric function
( 1F1(a,b;x) ) evaluated at the values a, b, and x.
- y = hyperu(a,b,x)
- hyperu(a,b,x) returns the confluent hypergeometric function of the
second kind U(a,b,x).
- (y,err) = hyp2f0(a,b,x,type)
- hyp2f0(a,b,x,type) returns (y,err) with the hypergeometric function 2F0 in y and an error estimate in err. The input type determines a convergence factor and
can be either 1 or 2.
- (y,err) = hyp1f2(a,b,c,x)
- hyp1f2(a,b,c,x) returns (y,err) with the hypergeometric function 1F2 in y and an error estimate in err.
- (y,err) = hyp3f0(a,b,c,x)
- hyp3f0(a,b,c,x) returns (y,err) with the hypergeometric function 3F0 in y and an error estimate in err.
Parabolic Cylinder Functions
- (d,dp) = pbdv(v,x)
- pbdv(v,x) returns (d,dp) with the parabolic cylinder function Dv(x) in
d and the derivative, Dv'(x) in dp.
- (v,vp) = pbvv(v,x)
- pbvv(v,x) returns (v,vp) with the parabolic cylinder function Vv(x) in
v and the derivative, Vv'(x) in vp.
- (w,wp) = pbwa(a,x)
- pbwa(a,x) returns (w,wp) with the parabolic cylinder function W(a,x) in
w and the derivative, W'(a,x) in wp. May not be accurate for large (>5)
arguments in a and/or x.
Legendre Functions
- y = lpmv(m,v,x)
- lpmv(m,v,x) returns the associated legendre function of integer order
m and nonnegative degree v: |x|<=1.
Mathieu Functions
- lmbda = mathieu_a(m,q)
- mathieu_a(m,q) returns the characteristic value for the even solution,
ce_m(z,q), of Mathieu's equation
- lmbda = mathieu_b(m,q)
- mathieu_b(m,q) returns the characteristic value for the odd solution,
se_m(z,q), of Mathieu's equation
- lmbda = mathieu_b(m,q)
- mathieu_b(m,q) returns the characteristic value for the odd solution,
se_m(z,q), of Mathieu's equation
- (y,yp) = mathieu_cem(m,q,x)
- mathieu_cem(m,q,x) returns the even Mathieu function, ce_m(x,q),
of order m and parameter q evaluated at x (given in degrees).
Also returns the derivative with respect to x of ce_m(x,q)
- (y,yp) = mathieu_sem(m,q,x)
- mathieu_sem(m,q,x) returns the odd Mathieu function, se_m(x,q),
of order m and parameter q evaluated at x (given in degrees).
Also returns the derivative with respect to x of se_m(x,q).
- (y,yp) = mathieu_modcem1(m,q,x)
- mathieu_modcem1(m,q,x) evaluates the even modified Matheiu function
of the first kind, Mc1m(x,q), and its derivative at x for order m and
parameter q.
- (y,yp) = mathieu_modcem2(m,q,x)
- mathieu_modcem2(m,q,x) evaluates the even modified Matheiu function
of the second kind, Mc2m(x,q), and its derivative at x (given in degrees)
for order m and parameter q.
- (y,yp) = mathieu_modsem1(m,q,x)
- mathieu_modsem1(m,q,x) evaluates the odd modified Matheiu function
of the first kind, Ms1m(x,q), and its derivative at x (given in degrees)
for order m and parameter q.
- (y,yp) = mathieu_modsem2(m,q,x)
- mathieu_modsem2(m,q,x) evaluates the odd modified Matheiu function
of the second kind, Ms2m(x,q), and its derivative at x (given in degrees)
for order m and parameter q.
Spheroidal Wave Functions
- (s,sp) = pro_ang1(m,n,c,x)
- pro_ang1(m,n,c,x) computes the prolate sheroidal angular function
of the first kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
- (s,sp) = pro_rad1(m,n,c,x)
- pro_rad1(m,n,c,x) computes the prolate sheroidal radial function
of the first kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
- (s,sp) = pro_rad2(m,n,c,x)
- pro_rad2(m,n,c,x) computes the prolate sheroidal radial function
of the second kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
- (s,sp) = obl_ang1(m,n,c,x)
- obl_ang1(m,n,c,x) computes the oblate sheroidal angular function
of the first kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
- (s,sp) = obl_rad1(m,n,c,x)
- obl_rad1(m,n,c,x) computes the oblate sheroidal radial function
of the first kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
- (s,sp) = obl_rad2(m,n,c,x)
- obl_rad2(m,n,c,x) computes the oblate sheroidal radial function
of the second kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0.
- cv = pro_cv(m,n,c)
- pro_cv(m,n,c) computes the characteristic value of prolate spheroidal
wave functions of order m,n (n>=m) and spheroidal parameter c.
- cv = obl_cv(m,n,c)
- obl_cv(m,n,c) computes the characteristic value of oblate spheroidal
wave functions of order m,n (n>=m) and spheroidal parameter c.
- (s,sp) = pro_ang1_cv(m,n,c,cv,x)
- pro_ang1_cv(m,n,c,cv,x) computes the prolate sheroidal angular function
of the first kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires pre-computed
characteristic value.
- (s,sp) = pro_rad1_cv(m,n,c,cv,x)
- pro_rad1_cv(m,n,c,cv,x) computes the prolate sheroidal radial function
of the first kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires pre-computed
characteristic value.
- (s,sp) = pro_rad2_cv(m,n,c,cv,x)
- pro_rad2_cv(m,n,c,cv,x) computes the prolate sheroidal radial function
of the second kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires pre-computed
characteristic value.
- (s,sp) = obl_ang1_cv(m,n,c,cv,x)
- obl_ang1_cv(m,n,c,cv,x) computes the oblate sheroidal angular function
of the first kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires pre-computed
characteristic value.
- (s,sp) = obl_rad1_cv(m,n,c,cv,x)
- obl_rad1_cv(m,n,c,cv,x) computes the oblate sheroidal radial function
of the first kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires pre-computed
characteristic value.
- (s,sp) = obl_rad2_cv(m,n,c,cv,x)
- obl_rad2_cv(m,n,c,cv,x) computes the oblate sheroidal radial function
of the second kind and its derivative (with respect to x) for mode paramters
m>=0 and n>=m, spheroidal parameter c and |x|<1.0. Requires pre-computed
characteristic value.
Struve Functions
- y = struve(v,x)
- struve(v,x) returns the Struve function Hv(x) of order v at x, x
must be positive unless v is an integer.
- y = modstruve(v,x)
- modstruve(v,x) returns the modified Struve function Lv(x) of order
v at x, x must be positive unless v is an integer and it is recommended
that |v|<=20.
- y = itstruve0(x)
- itstruve0(x) returns the integral of the Struve function of order 0
from 0 to x: integral(H0(t), t=0..x).
- y = it2struve0(x)
- it2struve0(x) returns the integral of the Struve function of order 0
divided by t from x to infinity: integral(H0(t)/t, t=x..inf).
- y = itmodstruve0(x)
- itmodstruve0(x) returns the integral of the modified Struve function
of order 0 from 0 to x: integral(L0(t), t=0..x).
Kelvin Functions
- (Be, Ke, Bep, Kep) = kelvin(x)
- kelvin(x) returns the tuple (Be, Ke, Bep, Kep) which containes
complex numbers representing the real and imaginary Kelvin functions
and their derivatives evaluated at x. For example,
kelvin(x)[0].real = ber x and kelvin(x)[0].imag = bei x with similar
relationships for ker and kei.
- y = ber(x)
- ber(x) returns the Kelvin function ber x
- y = bei(x)
- bei(x) returns the Kelvin function bei x
- y = ker(x)
- ker(x) returns the Kelvin function ker x
- y = kei(x)
- kei(x) returns the Kelvin function ker x
- y = berp(x)
- berp(x) returns the derivative of the Kelvin function ber x
- y = beip(x)
- beip(x) returns the derivative of the Kelvin function bei x
- y = kerp(x)
- kerp(x) returns the derivative of the Kelvin function ker x
- y = keip(x)
- keip(x) returns the derivative of the Kelvin function kei x
Other Special Functions
- y = expn(n,x)
- expn(n,x) returns the exponential integral for integer n and
non-negative x and n: integral(exp(-x*t) / t**n, t=1..inf).
- y = exp1(z)
- exp1(z) returns the exponential integral (n=1) of complex argument
z: integral(exp(-z*t)/t,t=1..inf).
- y = expi(z)
- expi(x) returns an exponential integral of argument x defined as
integral(exp(t)/t,t=-inf..x). See expn for a different exponential
integral.
- y = wofz(z)
- wofz(z) returns the value of the fadeeva function for complex argument
z: exp(-z**2)*erfc(-i*z)
- y = besselpoly(a,lam,nu)
- besselpoly(a,lam,nu) returns the value of the integral:
integral(x**lam * jv(nu,2*a*x),x=0..1).
- y = dawsn(x)
- dawsn(x) returns dawson's integral: exp(-x**2) *
integral(exp(t**2),t=0..x).
- (shi,chi) = shichi(x)
- shichi(x) returns the hyperbolic sine and cosine integrals:
integral(sinh(t)/t,t=0..x) and eul + ln x +
integral((cosh(t)-1)/t,t=0..x) where eul is Euler's Constant.
- (si,ci) = sici(x)
- sici(x) returns in si the integral of the sinc function from 0 to x:
integral(sin(t)/t,t=0..x). It returns in ci the cosine integral: eul + ln x +
integral((cos(t) - 1)/t,t=0..x).
- y = spence(x)
- spence(x) returns the dilogarithm integral: -integral(log t /
(t-1),t=1..x)
- y = zeta(x,q)
- zeta(x,q) returns the Riemann zeta function of two arguments:
sum((k+q)**(-x),k=0..inf)
- y = zetac(x)
- zetac(x) returns 1.0 - the Riemann zeta function: sum(k**(-x), k=2..inf)
Convenience Functions
- y = cbrt(x)
- cbrt(x) returns the real cube root of x.
- y = exp10(x)
- exp10(x) returns 10 raised to the x power.
- y = exp2(x)
- exp2(x) returns 2 raised to the x power.
- y = radian(d,m,s)
- radian(d,m,s) returns the angle given in (d)egrees, (m)inutes, and
(s)econds in radians.
- y = cosdg(x)
- cosdg(x) calculates the cosine of the angle x given in degrees.
- y = sindg(x)
- sindg(x) calculates the sine of the angle x given in degrees.
- y = tandg(x)
- tandg(x) calculates the tangent of the angle x given in degrees.
- y = cotdg(x)
- cotdg(x) calculates the cotangent of the angle x given in degrees.
- y = log1p(x)
- log1p(x) calculates log(1+x) for use when x is near zero.
- y = expm1(x)
- expm1(x) calculates exp(x) - 1 for use when x is near zero.
- y = cosm1(x)
- calculates cos(x) - 1 for use when x is near zero.
- y = round(x)
- Returns the nearest integer to x as a double precision
floating point result. If x ends in 0.5 exactly, the
nearest even integer is chosen.