Asymptotic behavior of the Beta function

Question

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 $|y_1|+|y_2|\to \infty$ (from below and above). I used the representation $$ B(z_1,z_2) = \frac{\Gamma(z_1)\Gamma(z_2)}{\Gamma(z_1+z_2)} $$ and the Stierling formula in the form $$ |\Gamma(x_j+iy_j)| \sim \sqrt{2\pi}\exp \biggl( -x-\frac \pi 2 |y|+(x-\frac 1 2)\ln |y_j|\biggr), \quad j=1,2, $$ to obtain an estimate $$ |B(z_1,z_2)| \sim \sqrt{2\pi} \frac{|y_1|^{x_1-1/2} |y_2|^{x_2-1/2}}{|y_1+y_2|^{x_1+x_2-1/2}} \exp\biggl( -\frac{\pi}{2}(|y_1|+|y_2|-|y_1+y_2|) \biggr), $$ but the latter estimate holds only when $|y_1|$, $|y_2|$, $|y_1+y_2|\to \infty$ and is useless in this case due to the argument of the exponent. Maybe there is some other way to obtain the desired estimate?