I'm looking for a theorem that will allow me to state,

[tex]

F(z) = \int_0^b \frac{f(\tau)}{\tau - z} d\tau \sim \int_0^b \frac{f(\tau)}{\tau} d\tau

[/tex]

as [itex]z \to 0[/itex] and [itex]f(\tau): \mathbb{C} \to \mathbb{C}[/itex], but it's

*real*along the path of integration.

I believe [itex]f(\tau)[/tex] satisfies these properties:

- [itex]f(0) = 0[/itex] and [itex]f(b) = 0[/itex]
- [itex]f(\tau)[/itex] is Holder continuous along the path of integration

Note that F(z) is well-defined and Holder continuous along the line because of the values of f at the endpoints.

An example would be,

[tex]

\int_0^1 \frac{\sqrt{\tau}}{\tau - z} d\tau

[/tex]