A Real Kaiser

Dear Real Kaiser:

Thanks for your response to my question about the inverse of sinc. The inverse we were looking for actually does not have to be a function. for example, for the function $f(x)=x^{2}$, its inverse $f^{-1}(x)=\pm {\sqrt {x}}$ is not a function, but is nonetheless a relationship. So here I am not looking for strictly a function, but just the inverse relationship.

Now you mentioned that the inverse relationship cannot be expressed in close form using elementary functions. I am just wondering why that is the case, is there any proof for this conclusion? And has anyone invented a special name for the relationship to denote the inverse of sinc if it turns out to be non expressible using elementary functions? For example, the area under the "bell-curve" is this integral: $\int _{-\infty }^{+\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {\left(x-\mu \right)^{2}}{2\sigma ^{2}}}}dx$ it is not expressible using close expression of elementary function, it can only be expressed using infinite series expansion. However, it still has a special name that mathematician assigns to it, the "erf(x)" function. So likewise I would like to know if the inverse of sinc, $sinc^{-1}(x)$, also has a special name in math.

But most importantly I would like to know if any mathematician had worked out a proof that the inverse of sinc cannot be expressed using closed expression of elementary function.

Thanks.

76.65.12.133 (talk) 03:14, 27 January 2008 (UTC)

Well, I don't know about a proof regarding how its inverse cannot be represented as a closed expression. And as for a name, historically, a name (and notation to refer to it again and again) is assigned to a function if that function is "useful" somehow in some application and if it pops up again and again somewhere. So nomenclature is assigned to make it easy to refer to it and write it everywhere. I don't think that the inverse of the sine cardinal function is very special so it is just called the inverse of the sine cardinal function. Its integral is important and named but not its inverse. I also think that its inverse can be found using power series expansion (one way is to write and solve a differential equation) but I am not sure what it would be. A Real Kaiser (talk) 05:47, 27 January 2008 (UTC)

Thanks a lot again! 76.65.12.133 (talk) 19:48, 27 January 2008 (UTC)

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