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application of sine integral at infinity
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(Application)
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For finding the value of the improper integral
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(1) |
we first use the partial fraction representation $$\frac{1}{x(1\!+\!x^2)} = \frac{1}{x}-\frac{x}{1\!+\!x^2}.$$ Thus we may write $$f(a) = \int_0^\infty\frac{\sin{ax}}{x}\,dx-\int_0^\infty\frac{x\sin{ax}}{1+x^2}\,dx.$$ But by the entry sine integral at infinity, the first integral equals $\displaystyle\frac{\pi}{2}$ . When we check $$f'(a) \;=\; \int_0^\infty\frac{\cos{ax}}{1\!+\!x^2}\,dx, \quad f''(a) \;=\; -\!\int_0^\infty\frac{x\sin{ax}}{1\!+\!x^2}\,dx,$$ we see that there is the linear
differential equation
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(2) |
i.e. $$f''-f \;=\; -\frac{\pi}{2},$$ satisfied by the sought function $a \mapsto f(a)$ . We have the initial conditions $$f(0) \;=\; \int_0^\infty{0}\,dx \;=\; 0, \quad f'(0) \;=\; \int_0^\infty\frac{dx}{1\!+\!x^2} \;=\; \sijoitus{a}{\quad b}\!\arctan{x} \;=\; \frac{\pi}{2}.$$ Therefore the general solution $$f(a) \;=\; C_1e^a+C_2e^{-a}+\frac{\pi}{2}$$ of (2) requires that $C_1 = 0$ , $C_2 = \frac{\pi}{2}$ , and consequently the sought integral $f(a)$ has the value
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(3) |
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"application of sine integral at infinity" is owned by pahio.
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generalisation of sine integral at infinity |
This object's parent.
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Cross-references: general solution, initial conditions, function, linear differential equation, integral, sine integral at infinity, improper integral
There is 1 reference to this entry.
This is version 2 of application of sine integral at infinity, born on 2009-01-23, modified 2009-01-24.
Object id is 11547, canonical name is ApplicationOfSineIntegralAtInfinity.
Accessed 835 times total.
Classification:
| AMS MSC: | 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems) | | | 26A36 (Real functions :: Functions of one variable :: Antidifferentiation) | | | 34A12 (Ordinary differential equations :: General theory :: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions) | | | 34A34 (Ordinary differential equations :: General theory :: Nonlinear equations and systems, general) |
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Pending Errata and Addenda
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