Difference between revisions of "Dirichlet discontinuous multiplier"
From Encyclopedia of Mathematics
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The integral | The integral | ||
| − | + | $$\int\limits_0^\infty\frac{\sin\alpha x}{x}\cos\beta xdx=\begin{cases}\frac\pi2&\text{if }\beta<\alpha,\\\frac\pi4&\text{if }\beta=\alpha,\\0&\text{if }\beta>\alpha,\end{cases}$$ | |
| − | which is a discontinuous function of the parameters | + | which is a discontinuous function of the parameters $\alpha>0$ and $\beta>0$. Used by P.G.L. Dirichlet in his studies on the attraction of ellipsoids [[#References|[1]]]. The integral was encountered earlier in the work of J. Fourier, S. Poisson and A. Legendre. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.G.L. Dirichlet, "Werke" , '''1''' , Chelsea, reprint (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.M. Fichtenholz, "Differential und Integralrechnung" , '''2–3''' , Deutsch. Verlag Wissenschaft. (1964)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.G.L. Dirichlet, "Werke" , '''1''' , Chelsea, reprint (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.M. Fichtenholz, "Differential und Integralrechnung" , '''2–3''' , Deutsch. Verlag Wissenschaft. (1964)</TD></TR></table> | ||
Revision as of 09:00, 6 September 2014
The integral
$$\int\limits_0^\infty\frac{\sin\alpha x}{x}\cos\beta xdx=\begin{cases}\frac\pi2&\text{if }\beta<\alpha,\\\frac\pi4&\text{if }\beta=\alpha,\\0&\text{if }\beta>\alpha,\end{cases}$$
which is a discontinuous function of the parameters $\alpha>0$ and $\beta>0$. Used by P.G.L. Dirichlet in his studies on the attraction of ellipsoids [1]. The integral was encountered earlier in the work of J. Fourier, S. Poisson and A. Legendre.
References
| [1] | P.G.L. Dirichlet, "Werke" , 1 , Chelsea, reprint (1969) |
| [2] | G.M. Fichtenholz, "Differential und Integralrechnung" , 2–3 , Deutsch. Verlag Wissenschaft. (1964) |
How to Cite This Entry:
Dirichlet discontinuous multiplier. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_discontinuous_multiplier&oldid=33248
Dirichlet discontinuous multiplier. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_discontinuous_multiplier&oldid=33248
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article