Description
I was wondering if we could add the values of the Sinc function at the complex infinities. In #14854, there was some talk to push this in a separate pull request. However, as @simonbyrne mentioned:
If we're going to fix it, we should also fix the cases where it returns incorrect NaNs for finite values as well (e.g. sinc(0.0 +250.0*im)).
However this is probably a bit of work: it will require a bit of complex analysis to figure out what those limits should be (unless you can find a good reference), and fix some of the other incorrect functions (e.g. sinpi(1+300*im), as well as adding tests and docs.
Although I am not quite sure how to remedy the incorrect NaNs for finite values. I have computed the complex limits for the Sinc function and found the following. Any checks or advice or collaboration would be greatly appreciated.
The Sinc function
Given that:
sinc(x+Iy) = [Fr(x,y) + I Fi(x,y)]/((x^2+y^2)*π),
where
Fr(x,y) = x*sinpi(x)*cosh(πy) + y*cospi(x)*sinh(πy),
and
Fi(x,y) = x*cospi(x)*sinh(πy) - y*sinpi(x)*cosh(πy).We have three cases to consider
- x is identically 0, in which case, we can simplify things to:
sinc(x+Iy) = sinh(πy)/(πy).- x is an integer,hence also a zero of sinpi(x) in which case, we can simplify things to:
sinc(x+Iy) = [Fr(x,y) + I Fi(x,y)]/((x^2+y^2)*π),
where
Fr(x,y) = y*cospi(x)*sinh(πy),
and
Fi(x,y) = x*cospi(x)*sinh(πy).- x+0.5 is an integer, hence also a zero of cospi(x) in which case, we can simplify things to:
sinc(x+Iy) = [Fr(x,y) + I Fi(x,y)]/((x^2+y^2)*π),
where
Fr(x,y) = x*sinpi(x)*cosh(πy),
and
Fi(x,y) = - y*sinpi(x)*cosh(πy).With this in mind, this gives rise to the following function definition:
function sinc(x::Complex)
xr, xi = reim(x)
if x == 0
return one(x)
elseif isinf(xr) && isfinite(xi) # Real part is infinite, Imaginary part is finite.
return zero(x)
elseif isfinite(xr) && isinf(xi) # Real part is finite, Imaginary part is infinite.
Si = sign(xi)
Sr = sign(xr)
Sc = sign(cospi(xr))
Ss = sign(sinpi(xr))
if xr == 0 # Real part is 0
return complex(Inf,0)
elseif isinteger(xr) # Real part is an integer, hence also a zero of sinpi(xr)
return oftype(x,Inf)*complex(Sc,Si*Sr*Sc)
elseif isinteger(xr+0.5) # Real part+0.5 is an integer, hence also a zero of cospi(xr)
return oftype(x,Inf)*complex(Sr*Ss,-Si*Ss)
else
return oftype(x,Inf)*complex(Sc,-Si*Ss)
end
elseif isinf(xr) && isinf(xi) # Real part and Imaginary part are infinite.
return oftype(x,NaN)*complex(1,1) # This limit does not exist. There are infinite number of paths.
else # Real part and Imaginary part are finite.
return oftype(x,sinpi(x)/(pi*x))
end
endLike I said, any help with this issue would be gladly appreciated! Thanks.
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