Regularization by differentiation respect to a parameter

[zetafunction]

let be the integrals

$$\int_{0}^{\infty}dxx^{2} (x-a)^{1/2}=I1$$ and

$$\int_{0}^{\infty}dxx^{2} (x-a)^{-1/2}=I2$$

is then correct that $$I2= 2\frac{dI1}{da}$$

whenever applying a regularization scheme , is it correct to differentiate with respct to external parameters ??

 

[A. Neumaier]

let be the integrals

$$\int_{0}^{\infty}dxx^{2} (x-a)^{1/2}=I1$$ and

$$\int_{0}^{\infty}dxx^{2} (x-a)^{-1/2}=I2$$

is then correct that $$I2= 2\frac{dI1}{da}$$

whenever applying a regularization scheme , is it correct to differentiate with respct to external parameters ??

Usually it is correct - typically one simply assumes an analytic dependence on a.

But rigorous proofs would have to take into account the detailed properties of a regularization scheme, and the meaning of the integrals...