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

Famous quotes containing the words relation, integral, imaginary and/or argument:

    “You must realize that I was suffering from love and I knew him as intimately as I knew my own image in a mirror. In other words, I knew him only in relation to myself.”
    —Angela Carter (1940–1992)

    “Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made me—a book consubstantial with its author, concerned with my own self, an integral part of my life; not concerned with some third-hand, extraneous purpose, like all other books.”
    —Michel de Montaigne (1533–1592)

    “All legendary obstacles lay between
    Us, the long imaginary plain,
    The monstrous ruck of mountains”
    —John Montague (b. 1929)

    “The argument ad feminam, all the old knives
    that have rusted in my back, I drive in yours,
    ma semblable, ma soeur!”
    —Adrienne Rich (b. 1929)