Airy Functions
- (Ai,Aip,Bi,Bip) = airy(x)
- airy(x) calculates the Airy functions and their derivatives
evaluated at x. 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.
Elliptic Functions and Integrals
- (sn,cn,dn,ph) = ellpj_sn(u,m)
- ellpj 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 = ellpe(m1)
- ellpe(m1) returns the complete integral of the second kind:
integral(sqrt(1-(1-m1)*sin(t)**2),t=0..pi/2)
- y = ellie(phi,m)
- ellie(phi,m) returns the incomplete elliptic integral of the
second kind: integral(sqrt(1-m*sin(t)**2),t=0..phi)
- y = ellpk()
- ellpk(m1) returns the complete integral of the first kind:
integral(1/sqrt(1-(1-m1)*sin(t)**2),t=0..pi/2)
- y = ellik(phi,m)
- ellik(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,x)
- jv(v,x) returns the Bessel function of real order v at x.
- y = yn(n,x)
- yn(n,x) returns the Bessel function of the second kind of integer
order n at x.
- y = yv(n,x)
- yv(v,x) returns the Bessel function of the second kind of real
order v at x.
- y = kn(n,x)
- kn(n,x) returns the modified Bessel function of the third kind for
integer order n at x.
- y = iv(v,x)
- iv(v,x) returns the modified Bessel function of real order v of
x. If x is negative, v must be integer valued.
- 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)
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)
- y = fdtr(df1,df2,x)
- fdtr(df1,df2,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 = (u1/df1)/(u2/df2), where u1 and
u2 are random variables having Chi square distributions with df1 and
df2 degrees of freedom, respectively.
- y = fdtrc(df1,df2,x)
- fdtrc(df1,df2,x) returns the area from x to infinity under the F
density function.
- x = fdtri(df1,df2,p)
- fdtri(df1,df2,p) finds the F density argument x such that the
integral from x to infinity of the F density is equal to the given
probability 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)
- y = gdtrc(a,b,x)
- gdtrc(a,b,x) returns the integral from x to infinity of the gamma
probability density function.
- 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) is equal
to 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) = igamc( k+1, m).
Arguments must both be positive and k an integer.
- y = pdtrc(k,m)
- pdtr(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) = igam( k+1, m).
Arguments must both be positive and k an integer.
- m = pdtri(k,y)
- pdtri(k,y) returns the Poisson variable x such that the integral
from 0 to x of the Poisson density is equal to the given probability
y: calculated by igami( k+1, y). k must be a nonnegative integer and
y between 0 and 1.
- p = stdtr(k,t)
- stdtr(k,t) returns the integral from minus infinity to t of the Student t
distribution with integer k > 0 degrees of freedom:
gamma((k+1)/2)/(sqrt(k*pi)*gamma(k/2)) *
integral((1+x**2/k)**(-k/2-1/2),x=-inf..t)
- t = stdtri(k,p)
- stdtri(k,p) returns the argument t such that stdtr(k,t) is equal to p.
- p = 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 Gaussian probability density
function, integrated from minus infinity to x:
1/sqrt(2*pi) * integral(exp(-t**2 / 2),t=-inf..x)
- y = ndtri(x)
- ndtri(x) 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 = erf(x)
- erf(x) returns the error function defined as
2/sqrt(pi) * integral(exp(-t**2 / 2),t=0..x)
- y = erfc(x)
- erfc(x) returns 1 - erf(x).
- y = smirnov(n,e)
- smirnov(n,e) returns the exact Smirnov statistic for a one-sided
test equal to the probability that the maximum difference between a
theoretical distribution and an empirical one based on n samples is
greater that e.
- e = smirnovi(n,y)
- smirnovi(n,y) returns e such that smirnov(n,e) = y.
- p = kolmogorov(y)
- kolmogorov(y) returns Kolmogorov's limiting distribution of a
two-sided test or probability that sqrt(n) * max deviation > y.
- y = kolmogorovi(p)
- kolmogorovi(p) returns y such that kolmogorov(y) = p
Gamma and Related Functions
- y = gamma(x)
- gamma(x) returns the gamma function of the argument. The gamma
function is often referred to as the generalized factorial since
x*gamma(x) = gamma(x+1) and gamma(n+1) = n! for nonnegative n.
- y = lgam(x)
- lgam(x) returns the base e logarithm of the absolute value of the
gamma function of x: ln(|gamma(x)|)
- y = igam(a,x)
- igam(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 = igamc(a,x)
- igamc(a,x) returns the complemented incomplete gamma integral
defined as 1 / gamma(a) * integral(exp(-t) * t**(a-1), t=x..inf) = 1 -
igam(a,x). Both arguments must be positive.
- x = igami(a,y)
- igami(a,y) returns x such that igamc(a,x) = y. k must be a
nonnegative integer.
- y = beta(a,b)
- beta(a,b) returns gamma(a) * gamma(b) / gamma(a+b)
- y = lbeta(x)
- lbeta(a,b) returns the natural logarithm of the absolute value of
beta: ln(|beta(x)|).
- y = incbet(a,b,x)
- incbet(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 = incbi(a,b,y)
- incbi(a,b,y) returns x such that incbet(a,b,x) = y.
- y = psi(x)
- psi(x) is the logarithmic derivative of the gamma function
evaluated at x.
- y = rgamma(x)
- rgamma(x) returns one divided by the gamma function of x.
HyperGeometric Functions
- y = hyp2f1(a,b,c,x)
- hyp2f1(a,b,c,x) returns the gauss hypergeometric function
( 2F1(a,b;c;x) ).
- y = hyperg(a,b,x)
- hyperg(a,b,x) returns the confluent hypergeometeric function
( 1F1(a,b;x) ) evaluated at the values a, b, and x.
- (y,err) = hyp2f0(a,b,x,type)
- hyp2f0 returns 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) = onef2(a,b,c,x)
- onef2 returns the hypergeometric function 1F2 in y and an error
estimate in err.
- (y,err) = threef0(a,b,c,x)
- threef0 returns the hypergeometric function 3F0 in y and an error
estimate in err.
Other Special Functions
- y = expn(n,x)
- expn(n,x) return the exponential integral for integer n and
non-negative x and n: integral(exp(-x*t) / t**n, t=1..inf).
- (ssa,cca) = fresnl(x)
- fresnl(x) returns the fresnel sin and cos integrals: integral(sin(pi/2
* t**2),t=0..x) and integral(cos(pi/2 * t**2),t=0..x).
- 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 = 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 = 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 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.