The Dirichlet integral defines a seminorm on \ mathcal { D } ( \ Omega ).
2.
The Dirichlet distribution and the Dirichlet process, based on the Dirichlet integral, are named after him.
3.
The first integral, the Dirichlet integral, is absolutely convergent for positive ? but only conditionally convergent when ? is 0.
4.
This paper gave the first rigorous proof of the convergence of Fourier series under fairly general conditions ( piecewise continuity and monotonicity ) by considering partial sums, which Dirichlet transformed into a particular Dirichlet integral involving what is now called the Dirichlet kernel.
5.
The random variable \ frac { \ sin ( x ) e ^ x } { x } has no expected value according to Lebesgue integration, but using conditional convergence and interpreting the integral as a Dirichlet integral, which is an improper Riemann integral, we can say:
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