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I am using a formula that results in finding the inverse cosine of a Sinc function, and am curious if such a function has a short form name for it, and perhaps any interesting properties? (Just as Sine(x)/x is defined as "Sinc(x)".

As in... The "Fancy Greek Letter Here" Function is defined as $cos^{-1}(Sinc(x/2)/\sqrt{2})$

And here I am using the normalized definition for Sinc as $Sinc(x)=\frac{Sin(\pi x)}{\pi x}$

The actual function I am using is of the form $cos^{-1}\left(\frac{\sqrt{2}sin(\pi x/2)}{\pi x}\right)$ .

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    $\begingroup$ 1) Your actual function is much different and 2) there is no such named function AFAIK. $\endgroup$ Commented Mar 8, 2017 at 21:51
  • $\begingroup$ It somewhat resembles the sine integral though. $\endgroup$
    – user856
    Commented Mar 8, 2017 at 21:54
  • $\begingroup$ @SimplyBeautifulArt Thanks I made the question match my actual case (after two edits). $\endgroup$ Commented Mar 8, 2017 at 21:54
  • $\begingroup$ @Rahul That is a very good observation- $cos^{-1}(sin(x)/x)$ does appear to be the sine integral at first glance. $\endgroup$ Commented Mar 8, 2017 at 22:11
  • $\begingroup$ May be, within a few years, it will be known as Boschen function $\endgroup$ Commented Mar 9, 2017 at 10:44

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I do not know what you plan to do with function $$f(x)=\cos ^{-1}\left(\frac{\sqrt{2} \sin \left(\frac{\pi x}{2}\right)}{\pi x}\right)$$ but, if you consider the range $-1\leq x \leq 1$, it is very well approximated by a Padé approximant $$f(x)\approx \frac{\frac{\pi }{4}+\frac{\pi ^2 (97+58 \pi ) }{2328}x^2+\frac{\pi ^4 (32424+4603 \pi ) }{11733120}x^4}{1+\frac{29 \pi ^2 }{291}x^2+\frac{4603 \pi ^4 }{2933280}x^4}$$ So, solving for $x$ equation $f(x)=a$ (with $\frac{\pi }{4}\leq a\leq \cos ^{-1}\left(\frac{\sqrt{2}}{\pi }\right)\approx 1.10385$) reduces to a quadratic in $x^2$.

For example, using $a=1$ would lead to $x=0.786935$ while the exact solution would be $x=0.786929$.

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