The functions available in the special package are universal functions, which follow broadcasting and automatic array looping.

Let us look at some of the most frequently used special functions −

• Cubic Root Function
• Exponential Function
• Relative Error Exponential Function
• Log Sum Exponential Function
• Lambert Function
• Permutations and Combinations Function
• Gamma Function

Let us now understand each of these functions in brief.

Cubic Root Function

The syntax of this cubic root function is – scipy.special.cbrt(x). This will fetch the element-wise cube root of x.

Let us consider the following example.

from scipy.special import cbrt
res = cbrt([10, 9, 0.1254, 234])
print res

The above program will generate the following output.

[ 2.15443469 2.08008382 0.50053277 6.16224015]

Exponential Function

The syntax of the exponential function is – scipy.special.exp10(x). This will compute 10**x element wise.

Let us consider the following example.

from scipy.special import exp10
res = exp10([2, 9])
print res

The above program will generate the following output.

[1.00000000e+02  1.00000000e+09]

Relative Error Exponential Function

The syntax for this function is – scipy.special.exprel(x). It generates the relative error exponential, (exp(x) – 1)/x.

When x is near zero, exp(x) is near 1, so the numerical calculation of exp(x) – 1 can suffer from catastrophic loss of precision. Then exprel(x) is implemented to avoid the loss of precision, which occurs when x is near zero.

Let us consider the following example.

from scipy.special import exprel
res = exprel([-0.25, -0.1, 0, 0.1, 0.25])
print res

The above program will generate the following output.

[0.88479687 0.95162582 1.   1.05170918 1.13610167]

Log Sum Exponential Function

The syntax for this function is – scipy.special.logsumexp(x). It helps to compute the log of the sum of exponentials of input elements.

Let us consider the following example.

from scipy.special import logsumexp
import numpy as np
a = np.arange(10)
res = logsumexp(a)
print res

The above program will generate the following output.

9.45862974443

Lambert Function

The syntax for this function is – scipy.special.lambertw(x). It is also called as the Lambert W function. The Lambert W function W(z) is defined as the inverse function of w * exp(w). In other words, the value of W(z) is such that z = W(z) * exp(W(z)) for any complex number z.

The Lambert W function is a multivalued function with infinitely many branches. Each branch gives a separate solution of the equation z = w exp(w). Here, the branches are indexed by the integer k.

Let us consider the following example. Here, the Lambert W function is the inverse of w exp(w).

from scipy.special import lambertw
w = lambertw(1)
print w
print w * np.exp(w)

The above program will generate the following output.

(0.56714329041+0j)
(1+0j)

Permutations & Combinations

Let us discuss permutations and combinations separately for understanding them clearly.

Combinations − The syntax for combinations function is – scipy.special.comb(N,k). Let us consider the following example −

from scipy.special import comb
res = comb(10, 3, exact = False,repetition=True)
print res

The above program will generate the following output.

220.0

Note − Array arguments are accepted only for exact = False case. If k > N, N < 0, or k < 0, then a 0 is returned.

Permutations − The syntax for combinations function is – scipy.special.perm(N,k). Permutations of N things taken k at a time, i.e., k-permutations of N. This is also known as “partial permutations”.

Let us consider the following example.

from scipy.special import perm
res = perm(10, 3, exact = True)
print res

The above program will generate the following output.

720

Gamma Function

The gamma function is often referred to as the generalized factorial since z*gamma(z) = gamma(z+1) and gamma(n+1) = n!, for a natural number ‘n’.

The syntax for combinations function is – scipy.special.gamma(x). Permutations of N things taken k at a time, i.e., k-permutations of N. This is also known as “partial permutations”.

The syntax for combinations function is – scipy.special.gamma(x). Permutations of N things taken k at a time, i.e., k-permutations of N. This is also known as “partial permutations”.

from scipy.special import gamma
res = gamma([0, 0.5, 1, 5])
print res

The above program will generate the following output.

[inf  1.77245385  1.  24.]