1
vote
0answers
44 views

Asymptotic behavior of the Beta function

Let $B(z_1,z_2)$ be the Beta function, $z_1 = x_1 + iy_1$, $z_2 = x_2 + i y_2$. Suppose that $x_1$, $x_2 > 0$. I want to estimate the behavior of $|B(x_1+iy_1,x_2+iy_2)|$ as $|y_1|+|y_2|\to \infty$ ...
0
votes
1answer
80 views

Is the hypergeometric function $F(5/4,3/4; 2, z)$ bounded on $(0,1]$

Consider the classical hypergeometric function $F(5/4,3/4; 2, z)$ for $z\in (0,1]$. Is this bounded by some real number (independent of $z$)? I'm aware of Euler's formula: $$F(5/4,3/4; 2, z) = ...
4
votes
2answers
227 views

Bound for the Legendre function of the second kind of degree $1/2$

Let $Q_{1/2}(u)$ be the Legendre function of the second kind of degree $1/2$. One can show that $Q_{1/2}(u) = O(u^{-3/2})$ as $u\to \infty$; see Equation 21 in Section 3.9.2 of Higher transcendental ...