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 with, relation, integral, imaginary and/or argument:

    “[Man’s] life consists in a relation with all things: stone, earth, trees, flowers, water, insects, fishes, birds, creatures, sun, rainbow, children, women, other men. But his greatest and final relation is with the sun.”
    —D.H. (David Herbert)

    “Skepticism is unbelief in cause and effect. A man does not see, that, as he eats, so he thinks: as he deals, so he is, and so he appears; he does not see that his son is the son of his thoughts and of his actions; that fortunes are not exceptions but fruits; that relation and connection are not somewhere and sometimes, but everywhere and always; no miscellany, no exemption, no anomaly,—but method, and an even web; and what comes out, that was put in.”
    —Ralph Waldo Emerson (1803–1882)

    “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)

    “I am giddy; expectation whirls me round.
    Th’ imaginary relish is so sweet
    That it enchants my sense.”
    —William Shakespeare (1564–1616)

    “You have such strong words at command, that they make the smallest argument seem formidable.”
    —George Eliot [Mary Ann (or Marian)