It is common knowledge that 1st order Macdonald function [tex]K_{1} (z)[/tex] has following asymptotic behavior if [tex]Re (z) > 0[/tex](adsbygoogle = window.adsbygoogle || []).push({});

[tex]z K_{1} (z) \rightarrow 1 \ \mathrm{if} \ z \rightarrow 0[/tex],

[tex]\sqrt{z} e^{z} K_{1} (z) \rightarrow \sqrt{\frac{\pi}{2}} \ \mathrm{if} \ z \rightarrow \infty[/tex]

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

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

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

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# Macdonald function evaluation

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