Example 1

The function $t^p(1-t)^q$ is not bounded on $]0, 1[$ when $p < 0$ or $q < 0$; by the logarithmic criterion of order 0 applied on a neighbourhood of the points 0 and 1 the integral $\int_{0}^{1} t^p(1-t)^q \, dt$ converges if and only if $p > -1$ and $q > -1$. If so, this integral is called the Eulerian integral of the first kind and is denoted by $\mathbf{B}(p+1, q+1)$ (cf. VII, p. 312).