Trigonometric Integral - Relation With The Exponential Integral of 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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