Incomplete gamma functions

Incomplete gamma functions#

Truncated exponential function, \(e_n(x)\)#

math53.expn(n, x)#

Returns \(\displaystyle e_n(x) = \sum_{k=0}^n \frac{x^k}{k!} = \frac{\Gamma(n+1, x)}{\Gamma(n+1)} e^x\), the truncated exponential sum function, for \(n>0\).

See also: Ehrhardt [297] (3.10.25).

https://mathworld.wolfram.com/ExponentialSumFunction.html

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Expn(2,3)
ereal('5.2359877559829887307E-1')
>>> ereal.Expn(4,13)
ereal('5.3518479027559984754E-1')

Relative exponential, \(\mathrm{exprel}_n(x)\)#

math53.expreln(n, x)#

Returns \(\displaystyle \mathrm{exprel}_n(x) = \frac{n!}{x^n} \left(e^x - \sum_{k=0}^{n-1} \frac{x^k}{k!} \right) = e^x x^{-n} \left(\Gamma(1+n) - n \Gamma(n,x) \right) = {}_1F_1(1, 1+n, x)\).

See also Ehrhardt [297] (3.10.10).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Expreln(3, 4)
ereal('5.2359877559829887307E-1')
>>> ereal.Expreln(3, 12)
ereal('5.3518479027559984754E-1')
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