Exponential integrals, and related functions#
Entire cosine integral \(\mathrm{Cin}(x)\)#
- math53.cin(z)#
Returns the entire cosine integral of \(\displaystyle \mathrm{Cin}(x) = \int_0^{x} \frac{1-\cos(t)}{t} \, \mathrm{d}t\).
See also Wikipedia [1426], MathWorld [1046], NIST [849], Ehrhardt [309] (3.4.3).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.Cin(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Cin('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.Cin(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Cin('0.51') Gpr('5.3518479027559984754E-1')
Entire hyperbolic cosine integral \(\mathrm{Cinh}(x)\)#
- math53.cinh(z)#
Returns the entire hyperbolic cosine integral \(\displaystyle \mathrm{Cinh}(x) = \int_0^{x} \frac{\cosh(t)-1}{t} \, \mathrm{d}t\).
See also Wikipedia [1432], MathWorld [1058], NIST [846], Ehrhardt [309] (3.4.4).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.Cinh(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Cinh('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.Cinh(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Cinh('0.51') Gpr('5.3518479027559984754E-1')
Entire exponential integral \(\displaystyle \mathrm{Ein}(x)\)#
- math53.ein(x)#
Returns the entire exponential integral \(\displaystyle \mathrm{Ein}(x) = -\int_0^{x} \frac{1-e^{-t}}{t} \, \mathrm{d}t\).
See also Wikipedia [1432], MathWorld [1058], NIST [846], Ehrhardt [309] (3.4.11).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.Ein(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Ein('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.Ein(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Ein('0.51') Gpr('5.3518479027559984754E-1')
Scaled exponential integral \(\mathrm{e1s}(x)\)#
- math53.exp_integral_e1_scaled(z)#
Returns the scaled exponential integral \(\mathrm{e1s}(x) = e ^{-x} E_1(x)\) for \(x \ne 0\).
See also Wikipedia [1427], MathWorld [1052], NIST [844], BoostMath [160], Ehrhardt [309] (3.4.6).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.E1s(0.5) xreal('5.2359877559829887307E-1') >>> xreal.E1s('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.E1s(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.E1s('0.51') Gpr('5.3518479027559984754E-1')
Scaled exponential integral \(\mathrm{eis}(x)\)#
- math53.exp_integral_ei_scaled(x)#
Returns the scaled exponential integral \(\mathrm{eis}(x) = e ^{-x} \mathrm{Ei}(x)\) for \(x \ne 0\).
See also Wikipedia [1427], MathWorld [1052], NIST [844], BoostMath [160], Ehrhardt [309] (3.4.8).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.Eis(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Eis('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.Eis(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Eis('0.51') Gpr('5.3518479027559984754E-1')
Scaled exponential integral \(\mathrm{eisx2}(x)\)#
- math53.eisx2(x)#
Returns the scaled exponential integral \(\mathrm{eisx2}(x) = e ^{-x^2} \mathrm{Ei}(x^2)\) for \(x \ne 0\).
See also Wikipedia [1427], MathWorld [1052], NIST [844], BoostMath [160], Ehrhardt [309] (3.4.9).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.Eisx2(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Eisx2('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.Eisx2(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Eisx2('0.51') Gpr('5.3518479027559984754E-1')
Inverse of the exponential integral, \(\mathrm{Ei}^{-1}(x)\)#
- math53.ei_inv(x)#
Returns \(\mathrm{Ei}^{-1}(x) = \log(\mathrm{li}^{-1}(x))\), the functional inverse of the exponential integral, i.e. \(\mathrm{Ei}(\mathrm{Ei}^{-1}(x)) = x\).
See also Wikipedia [1427], MathWorld [1052], NIST [844], BoostMath [160], Ehrhardt [309] (3.4.10).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.EiInv(0.5) xreal('5.2359877559829887307E-1') >>> xreal.EiInv('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.EiInv(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.EiInv('0.51') Gpr('5.3518479027559984754E-1')
Generalized exponential integral \(E_p(p,x)\)#
- math53.gei(p, x)#
Returns the generalized exponential integral of real order \(\displaystyle E_p(x) = \int_{1}^{\infty} \frac{e^{-xt}}{t^p} \, \mathrm{d}t = x^{p-1} \Gamma(1-p, x) = x^{p-1} Q(1-p, x) \Gamma(1-p)\).
See also Wikipedia [1427], MathWorld [1053], NIST [485], BoostMath [161], Ehrhardt [309] (3.4.13).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.Gei(3.1, 0.5) xreal('5.2359877559829887307E-1') >>> xreal.Gei(3.1, '0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.Gei(3.1, 0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Gei(3.1, '0.51') Gpr('5.3518479027559984754E-1')
Exponential integral \(\displaystyle \beta_n(x)\)#
- math53.eibeta(n, x)#
Returns the exponential integral \(\displaystyle \beta_n(x) = \int_{1}^{1} t^n e^{-xt} \, \mathrm{d}t = x^{-n-1}\left(\Gamma(n+1,-x)-\Gamma(n+1,x)\right), n \ge 0\).
See also Abramowitz and Stegun. [4] (5.1.6) and (5.1.47), and Ehrhardt [309] (3.4.14).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.Eibeta(3, 0.5) xreal('5.2359877559829887307E-1') >>> xreal.Eibeta(3, '0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.Eibeta(3, 0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Eibeta(3, '0.51') Gpr('5.3518479027559984754E-1')
Inverse of the logarithmic integral, \(\mathrm{li}^{-1}(x)\)#
- math53.log_integral_inv(x)#
Returns \(\mathrm{li}^{-1}(x)\), the functional inverse of the logarithmic integral, i.e. \(\mathrm{li}(\mathrm{li}^{-1}(x)) = x\).
See also Wikipedia [1436], MathWorld [1063], NIST [847], Ehrhardt [309] (3.4.16).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.LiInv(0.5) xreal('5.2359877559829887307E-1') >>> xreal.LiInv('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.LiInv(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.LiInv('0.51') Gpr('5.3518479027559984754E-1')
Shifted sine integral \(\mathrm{si}(x) = \mathrm{Si}(x) - \pi/2\)#
- math53.shifted_sin_integral(x)#
Returns the shifted sine integral, \(\mathrm{si}(x) = \mathrm{Si}(x) - \pi/2\).
See also Wikipedia [1441], MathWorld [1065], NIST [849], Ehrhardt [309] (3.4.19).
An example in Python
>>> from xlcalcnet import xreal >>> xreal.ShiftedSi(0.5) xreal('5.2359877559829887307E-1') >>> xreal.ShiftedSi('0.51') xreal('5.3518479027559984754E-1')
An example in Visual Basic
>>> from xlcalcnet import Gpr >>> Gpr.ShiftedSi(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.ShiftedSi('0.51') Gpr('5.3518479027559984754E-1')