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 know there are no secrets in America. It’s quite different in England, where people think of a secret as a shared relation between two people.”
    —W.H. (Wystan Hugh)

    “... no one who has not been an integral part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.”
    —Angelina Grimké (1805–1879)

    “Girls are apt to imagine noble and enchanting and totally imaginary figures in their own minds; they have fanciful extravagant ideas about men, and sentiment, and life; and then they innocently endow somebody or other with all the perfections for their daydreams, and put their trust in him.”
    —Honoré De Balzac (1799–1850)

    “Of course poets have morals and manners of their own, and custom is no argument with them.”
    —Thomas Hardy (1840–1928)