i.2.31 Integrals depending on a parameter
condition: Examine the improper integral for convergence. Indicate at what values of the positive parameter \( \alpha \) the integral converges. \[ \iint_{00}^{\infty} \frac{d x d y}{\exp \left(x^{2}+y^{\alpha}\right)-1} \]
Integrals depending on a parameter, often referred to as parameter-dependent integrals or definite integrals with parameters, involve the integration of a function where one or more parameters are present. These parameter-dependent integrals can lead to a variety of problems and challenges. Here you can find problems related to convergence and divergence, calculation, limits, parametric integration and differentiation of parameter-dependent integrals. Also, there are usage and properties of Euler, Frullani, Poisson, Laplace, and Dirichlet integrals.