# Macdonald function evaluation

1. Apr 26, 2008

### Mr.Slava

It is common knowledge that 1st order Macdonald function $$K_{1} (z)$$ has following asymptotic behavior if $$Re (z) > 0$$

$$z K_{1} (z) \rightarrow 1 \ \mathrm{if} \ z \rightarrow 0$$,
$$\sqrt{z} e^{z} K_{1} (z) \rightarrow \sqrt{\frac{\pi}{2}} \ \mathrm{if} \ z \rightarrow \infty$$

I am interesting in a question, how to find following constraint from only asymptotic relations which are indicated above

$$|K_{1} (z)| \leq \mathrm{const} \cdot (|z|^{ -1/2 } + |z|^{ -1 }) e^{ - Re (z) }$$

( I cannot find an explanation in the 'H.Bateman, Higher Transcendental Functions'. Thanks)

Last edited: Apr 26, 2008