Lobachevskian formulas

The Lobachevskian formulas are two mathematical formulas for improper integrals in connection with the cardinal sine , which are to be assigned to the sub-area of analysis . According to the presentation by GM Fichtenholz in Volume II of the three-volume differential and integral calculus , they were found by the Russian mathematician Nikolai Ivanovich Lobachevsky (1792-1856).

Representation of the formulas

They are:

A real function is given

${\displaystyle f\colon [0,\infty [\to \mathbb {R} }$

with the following properties:

(1) is actually or improperly Riemann integrable in the interval . ${\displaystyle f}$ ${\displaystyle \left[0,{\frac {\pi }{2}}\right]}$

(2) The product function formed with the cardinal sine is improperly Riemann integrable in the interval . ${\displaystyle f\cdot \operatorname {si} \colon [0,\infty [\to \mathbb {R} \;,x\mapsto f(x)\cdot \operatorname {si} (x)}$ ${\displaystyle [0,\infty [}$

(3) is a - periodic function , so it always satisfies the equation . ${\displaystyle f}$ ${\displaystyle \pi }$ ${\displaystyle x\in [0,\infty [}$ ${\displaystyle f(x+{\pi })=f(x)}$

(4) always satisfies the equation . ${\displaystyle f}$ ${\displaystyle x\in [0,\pi ]}$ ${\displaystyle f({\pi }-x)=f(x)}$

Then:

(a) ${\displaystyle \int _{0}^{\infty }{f(x)\cdot {\frac {\sin x}{x}}}\,\mathrm {d} x=\int _{0}^{\frac {\pi }{2}}{f(x)}\,\mathrm {d} x}$

(b) ${\displaystyle \int _{0}^{\infty }{f(x)\cdot {\frac {{\sin }^{2}x}{x^{2}}}}\,\mathrm {d} x=\int _{0}^{\frac {\pi }{2}}{f(x)}\,\mathrm {d} x}$

Applications

With the help of the Lobachevskian formulas (and with the help of the usual calculation methods of integral calculus ) several identities can be derived, including the following:

(A-1) ${\displaystyle \int _{0}^{\infty }{\frac {\sin x}{x}}\,\mathrm {d} x={\frac {\pi }{2}}}$

(A-2) ${\displaystyle \int _{0}^{\infty }{\frac {{\sin }^{2n+1}x}{x}}\,\mathrm {d} x={\frac {\pi }{2}}\cdot {\frac {(2n-1)!!}{(2n)!!}}\;(n=0,1,2,3,\ldots )}$

(A-3) ${\displaystyle \int _{0}^{\infty }{\frac {{\sin }^{2}x}{x^{2}}}\,\mathrm {d} x={\frac {\pi }{2}}}$

(A-4) ${\displaystyle \int _{0}^{\infty }{\frac {\arctan {\left(a\cdot \sin x\right)}}{x}}\,\mathrm {d} x={\frac {\pi }{2}}\cdot \ln {\left(a+{\sqrt {1+a^{2}}}\right)}\;(a>0)}$

(A-5) ${\displaystyle \int _{0}^{\infty }{\ln {|{\sin x}|}\cdot {\frac {\sin x}{x}}}\,\mathrm {d} x=-{{\frac {\pi }{2}}\cdot \ln 2}}$

(A-6) ${\displaystyle \int _{0}^{\infty }{\frac {\ln {|{\cos x}|}}{x^{2}}}\,\mathrm {d} x=-{\frac {\pi }{2}}}$

(A-7) ${\displaystyle \int _{0}^{\infty }{\frac {{\ln }^{2}{|{\cos x}|}}{x^{2}}}\,\mathrm {d} x=-{\pi \cdot \ln 2}}$

Background: Partial fraction decomposition

As Fichtenholz explains, the Lobachevskian formulas are essentially based on the partial fraction decomposition of the two functions . The following applies here: ${\displaystyle x\mapsto {\frac {1}{\sin x}}\;,\;x\mapsto {\frac {1}{{\sin }^{2}x}}\;(x\notin {\pi \cdot \mathbb {Z} })}$

${\displaystyle {\frac {1}{\sin x}}={\frac {1}{x}}+\sum _{n=1}^{\infty }{(-1)^{n}\left({\frac {1}{x-n\pi }}+{\frac {1}{x+n\pi }}\right)}={\frac {1}{x}}+\sum _{n=1}^{\infty }{(-1)^{n}{\frac {2x}{x^{2}-n^{2}{\pi }^{2}}}}}$

such as

${\displaystyle {\frac {1}{{\sin }^{2}x}}={\frac {1}{x^{2}}}+\sum _{n=1}^{\infty }{\left({\frac {1}{\left[x-n\pi \right]^{2}}}+{\frac {1}{\left[x+n\pi \right]^{2}}}\right)}}$   .

literature

• GM Fichtenholz: Differential and Integral Calculus II . Translation from Russian and scientific editing: Dipl.-Math. Brigitte Mai, Dipl.-Math. Walter Mai (=  university books for mathematics . Volume 62 ). 6th edition. VEB Deutscher Verlag der Wissenschaften , Berlin 1974.

Individual evidence

1. ↑ GM Fichtenholz: Differential- und Integralrechner II. 1974, pp. 635–636, 655–657, 695, 832
2. ↑ Fichtenholz, op. Cit., Pp. 655-657, 695
3. ↑ Fichtenholz, op. Cit., Pp. 635–636
4. ↑ Fichtenholz, op.cit., P. 656
5. ↑ The double exclamation point that is double factorial function characterized.
6. ↑ Fichtenholz, op. Cit., Pp. 656–657
7. ↑ Fichtenholz, op.cit., Pp. 656, 697
8. ↑ ^{a b c} spruce wood, op.cit., P. 695
9. ↑ With that is the absolute value function characterized. ${\displaystyle |\cdot |}$
10. ↑ Fichtenholz, op.cit., Pp. 489, 656