I thought Sin(Theta)=Cos(Theta+pi/2) or something.... Its been 30 years since I did maths and I never was much good. I did pass though. Thanks for the definition of a Bessel function, I suppose you could have Sin of Tan and Tan of Tan too. I suppose they are called something else like Cherbochev polynomials, which I never did understand. I suppose that to get the Int(Cos(sin(theta))) d theta - or whatever the full expression is, you have to expand Cos(x) where x=Sin(theta) in terms of the infinite series for Cos(x) then expand each term of Cos(x) as powers of sin(theta) then expand Sin(theta) inside each term then integrate with respect to theta. Group into powers of theta and sum to get the first few terms. Then guess the general term somehow. Then do a numerical calculation to make sense of it.... ? BTW I think z here is a different z....? (from the pinhole). Chris Web Page http://www.chrisweb.pwp.blueyonder.co.uk/ |> -----Original Message----- |> From: owner-photoforum@listserver.isc.rit.edu |> [mailto:owner-photoforum@listserver.isc.rit.edu]On Behalf Of Bob Blakely |> Sent: 24 August 2003 08:19 |> To: List for Photo/Imaging Educators - Professionals - Students |> Subject: Re: Minimizing pinhole image falloff - Correction |> |> |> Damn close there! It's the integral of the cos of a sin. |> |> As best I can do with plain text... |> |> Jn(z) = 1 / pi * Integral, 0 to pi of cos[z * sin(t) - n * t] * dt |> |> where t is theta. |> |> Regards, |> Bob... |> -------------------------------------------- |> "Do not suppose that abuses are eliminated by destroying |> the object which is abused. Men can go wrong with wine |> and women. Shall we then prohibit and abolish women?" |> -Martin Luther |> |> ----- Original Message ----- |> From: "Chris" <nimbo@ukonline.co.uk> |> To: "List for Photo/Imaging Educators - Professionals - Students" |> <photoforum@listserver.isc.rit.edu> |> Sent: Saturday, August 23, 2003 2:01 PM |> Subject: RE: Minimizing pinhole image falloff - Correction |> |> |> > Thank you. |> > |> > We didn't do Bessel functions except in fm radio theory. The |> theory that |> I |> > don't remember involved an approximation. When I tried to do the |> > integration myself just now I got the integral of the sin of a |> sin which I |> > think is the differential form of the Bessel function. Very |> interesting. |> |> |>
