Integral sine

Course of the integral sine in the range 0 ≤ x ≤ 8π

The integral sine is a term from mathematics and refers to a by a integral given function . Joseph Liouville (1809–1882) proved that the cardinal sine cannot be integrated in an elementary way .

The integral sine is defined as the integral of the sinc function:

${\displaystyle \operatorname {Si} (x):=\int _{0}^{x}\operatorname {si} (t)\,\mathrm {d} t=\int _{0}^{x}{\frac {\sin t}{t}}\,\mathrm {d} t}$.

properties

• The integral can be evaluated at the limit crossing . The following applies: ${\displaystyle \lim _{x\to \infty }\mathrm {Si} \left(x\right)}$

${\displaystyle \lim _{x\to \infty }\mathrm {Si} \left(x\right)={\frac {\pi }{2}}}$

• Analogous to the complex Euler formula definition of the sine:

${\displaystyle \sin x={\frac {1}{2\mathrm {i} }}\left(\mathrm {e} ^{\mathrm {i} x}-\mathrm {e} ^{-\mathrm {i} x}\right)}$

holds with the integral exponential function ${\displaystyle \mathrm {Ei} }$

${\displaystyle \mathrm {Si} (x)={\frac {1}{2\mathrm {i} }}\left(\mathrm {Ei} (\mathrm {i} \ x)-\mathrm {Ei} (-\mathrm {i} \ x)\right)+{\frac {\pi }{2}}}$

• The expansion into a Taylor series at the point 0 gives the compact convergent series:

${\displaystyle \mathrm {Si} \left(x\right)=x-{\frac {x^{3}}{3!\cdot 3}}+{\frac {x^{5}}{5!\cdot 5}}-{\frac {x^{7}}{7!\cdot 7}}+\cdots \quad =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)!\cdot (2k+1)}}x^{2k+1}}$

Closely related is the integral cosine Ci (x), which together with the integral sine Si (x) forms a clothoid in parametric representation .

Special values

${\displaystyle \mathrm {Si} (\pi )=1{,}851937...}$ Wilbraham – Gibbs constant

Related limit values

${\displaystyle \lim _{x\to \infty }\int _{0}^{x}{\frac {\sin ^{2}t}{t^{2}}}\,dt={\frac {\pi }{2}}}$

 

${\displaystyle \lim _{x\to \infty }\int _{0}^{x}{\frac {\sin ^{3}t}{t^{3}}}\,dt={\frac {3}{8}}\pi }$

 

${\displaystyle \lim _{x\to \infty }\int _{0}^{x}{\frac {\sin ^{4}t}{t^{4}}}\,dt={\frac {\pi }{3}}}$

 

${\displaystyle \lim _{x\to \infty }\int _{0}^{x}{\frac {\sin ^{6}t}{t^{6}}}\,dt={\frac {11}{40}}\pi }$

See also

• Integral exponential function
• Integral cosine

literature

• Horst Nasert: About the general integral sine and integral cosine .
• Erwin O. Kreyszig (speaker: Alwin [Oswald] Walther ; co- speaker: Curt [Otto Walther] Schmieden ): About the general integral sine . ${\displaystyle \mathrm {Si} (z,a)}$Extract from inaugural dissertation, Institute for Practical Mathematics at the Technical University of Darmstadt.

Web links

• Eric W. Weisstein : Sine Integral . In: MathWorld (English).

Individual evidence

1. ↑ J. Liouville: “Mémoire. Sur la classification des Transcendantes et sur l'impossibilité d'exprimer les racines des certaines équations en fonction finie explicite des coefficients. Part 1 " . Journal de Mathématiques Pures et Appliquées , 2, 56-105, 1837.
2. ^ J. Liouville: “Suite du Mémoire. Sur la classification des Transcendantes et sur l'impossibilité d'exprimer les racines des certaines équations en fonction finie explicite des coefficients. Part 2 " . Journal de Mathématiques Pures et Appliquées, 3, 523-547, 1838.
3. ↑ J. Liouville: “Mémoire. Sur l'integration d'une classe d'Équations différentielles du second ordre en quantités finies explicites ” . Journal de Mathématiques Pures et Appliquées, 4, 423–456, 1839.
4. ^ Joseph (Fels) Ritt : Integration in Finite Terms: Liouville's Theory of Elementary Methods. Columbia University Press, New York 1948.
5. ^ Siegfried (Johannes) Gottwald : Handbook of Mathematics. A guide for school and practice, particularly suitable for self-study. Buch und Zeit Verlagsgesellschaft, Cologne 1986. ISBN 3-8166-0015-8 . P. 517 (704 pp.).
6. ↑ Eric W. Weisstein : Wilbraham-Gibbs Constant . In: MathWorld (English).