Absolite convergence of the beta function

[Ted123]

Determine the values of the complex parameters p and q for which the beta function \int_0^1 t^{p-1} (1-t)^{q-1}dt converges absolutely.

The solution says:

Split the integral into 2 parts: \displaystyle \int_0^1 t^{p-1} (1-t)^{q-1}dt = \int_0^{1/2} t^{p-1} (1-t)^{q-1}dt + \int_{1/2}^1 t^{p-1} (1-t)^{q-1}dt

As t \to 0 |t^{p-1} (1-t)^{q-1}| \sim |t^{p-1}| \Rightarrow \text{Re}(p)>0 As t\to 1 |t^{p-1} (1-t)^{q-1}| \sim |(1-t)^{q-1}| \Rightarrow \text{Re}(q)>0

Why are these implications true?

 

[sunjin09]

Determine the values of the complex parameters p and q for which the beta function \int_0^1 t^{p-1} (1-t)^{q-1}dt converges absolutely.

The solution says:

Split the integral into 2 parts: \displaystyle \int_0^1 t^{p-1} (1-t)^{q-1}dt = \int_0^{1/2} t^{p-1} (1-t)^{q-1}dt + \int_{1/2}^1 t^{p-1} (1-t)^{q-1}dt

As t \to 0 |t^{p-1} (1-t)^{q-1}| \sim |t^{p-1}| \Rightarrow \text{Re}(p)>0 As t\to 1 |t^{p-1} (1-t)^{q-1}| \sim |(1-t)^{q-1}| \Rightarrow \text{Re}(q)>0

Why are these implications true?

because otherwise a neighborhood of the endpoints is non-integrable?