Trigonometric Integral - Relation With The Exponential Integral of Imaginary Argument | Relation Exponential Integral Imaginary Argument

Relation With The Exponential Integral of Imaginary Argument

The function

is called the exponential integral. It is closely related to Si and Ci:

${\rm E}_1( {\rm i}\!~ x) = i\left(-\frac{\pi}{2} + {\rm Si}(x)\right)-{\rm Ci}(x) = i~{\rm si}(x) - {\rm ci}(x) \qquad(x>0)$

As each involved function is analytic except the cut at negative values of the argument, the area of validity of the relation should be extended to . (Out of this range, additional terms which are integer factors of appear in the expression).

Cases of imaginary argument of the generalized integro-exponential function are

$\int_1^\infty \cos(ax)\frac{\ln x}{x} \, dx = -\frac{\pi^2}{24}+\gamma\left(\frac{\gamma}{2}+\ln a\right)+\frac{\ln^2a}{2} +\sum_{n\ge 1}\frac{(-a^2)^n}{(2n)!(2n)^2},$

which is the real part of

$\int_1^\infty e^{iax}\frac{\ln x}{x} \, dx = -\frac{\pi^2}{24} + \gamma\left(\frac{\gamma}{2}+\ln a\right)+\frac{\ln^2 a}{2}-\frac{\pi}{2}i(\gamma+\ln a) + \sum_{n\ge 1}\frac{(ia)^n}{n!n^2}.$

Similarly

$\int_1^\infty e^{iax}\frac{\ln x}{x^2}dx =1+ia[-\frac{\pi^2}{24}+\gamma\left(\frac{\gamma}{2}+\ln a-1\right)+\frac{\ln^2 a}{2}-\ln a+1 -\frac{i\pi}{2}(\gamma+\ln a-1)]+\sum_{n\ge 1}\frac{(ia)^{n+1}}{(n+1)!n^2}.$

Read more about this topic: Trigonometric Integral

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