Domain of validity of an integral

[tourjete]

Homework Statement

This integral is actually part of a question I had on an exam about analytic continuation. The integral is
$$\int_0^1 dx \frac{x^n}{\sqrt{x^3+5}}$$. The first part of the question is "what is its domain of validity for absolute convergence of the integral?" It then goes on to ask for a valid analytic continuation (I know how to do that part)

Homework Equations

none

The Attempt at a Solution

The answer key my professor provided states that "the integrand only blows up at x=0, if at all, in a finite range of integration. As x -> 0, the integrand is O($x^n$), so there is absolute convergence if Real n > -1.

I can easily how how this is the domain of validity assuming the integral blows up at 0, but I can't see how the integral would possibly blow up at 0. Plugging in zero for x gives $\frac{0}{\sqrt{5}}$. I've tried putting the integral in other forms and I still don't see it. I taylor expanded the square root around x = 0 but that still doesn't cause it to blow up, nor does multiplying the top and bottom of the integral by the square root. I'm sure I'm just making some stupid calculus mistake somewhere...

 

[Ray Vickson]

Homework Statement

This integral is actually part of a question I had on an exam about analytic continuation. The integral is
$$\int_0^1 dx \frac{x^n}{\sqrt{x^3+5}}$$. The first part of the question is "what is its domain of validity for absolute convergence of the integral?" It then goes on to ask for a valid analytic continuation (I know how to do that part)

Homework Equations

none

The Attempt at a Solution

The answer key my professor provided states that "the integrand only blows up at x=0, if at all, in a finite range of integration. As x -> 0, the integrand is O($x^n$), so there is absolute convergence if Real n > -1.

I can easily how how this is the domain of validity assuming the integral blows up at 0, but I can't see how the integral would possibly blow up at 0. Plugging in zero for x gives $\frac{0}{\sqrt{5}}$. I've tried putting the integral in other forms and I still don't see it. I taylor expanded the square root around x = 0 but that still doesn't cause it to blow up, nor does multiplying the top and bottom of the integral by the square root. I'm sure I'm just making some stupid calculus mistake somewhere...

Did you not read the restriction Re(n) > -1? What happens if you take Re(n) = -1 (for example, n = -1)? What happens if you take Re(n) < -1, for example, n = -2?