How Can the Cauchy Integral Transform Be Defined to Avoid Singularities?

[tpm]

let be the next linear integral transform:

$$g_{k} (x)= \int_{-\infty}^{\infty}dt \frac{f(t)}{(t-x)^{k}}$$

no matter what f(t) is there is a singularity at the points where t=x how could you define it so it's finite avoiding the poles at t=x where k is a positive integer.

 

[AiRAVATA]

no matter what f(t) is there is a singularity at the points where t=x

False. When f(t) has a zero of order $\ge k$ in x, i.e. $f(t)=(t-x)^k f_0(t)$, and $f_0(t)\in L^1(0,\infty)$, such integral is well defined.

Another way the integral can be well defined is verified with complex variable and residue theory.