I was wondering if anyone is able to either prove or disprove the following identity. Please see attached PDF file.
This identity came up in my work recently. I won't go into the details of the application, but I did two different approximate derivations of a function I needed in an engineering application. The two separate functions appeared to give the same numerical answer, but look quite different. One funtion was very simple (y=0.5*x) and the other is much more complicated with an arctan function (see attachment).
My first assumption was that these are probably exactly equal functions; however, when I numerically evaluate the error using double precision calculations in Matlab, I find the error near x=pi/3 to be considerable more than what I expect with double precision. The error is much less as x moves away from this value. Now this error could be due to round off error accumulation, but I'm not convinced of that. (Actually, it probably is due to a zero/zero condition)
I wanted to either prove or disprove this identity, but I've been too busy and can't justify doing this as part of my work since it's just a point of curiousity.
I thought someone here might enjoy tackling this one. I suspect the identity is true, and there is probably a simple elegant way to prove it. I thought of doing out a Taylor expansion, but that looks tedious.
This identity came up in my work recently. I won't go into the details of the application, but I did two different approximate derivations of a function I needed in an engineering application. The two separate functions appeared to give the same numerical answer, but look quite different. One funtion was very simple (y=0.5*x) and the other is much more complicated with an arctan function (see attachment).
My first assumption was that these are probably exactly equal functions; however, when I numerically evaluate the error using double precision calculations in Matlab, I find the error near x=pi/3 to be considerable more than what I expect with double precision. The error is much less as x moves away from this value. Now this error could be due to round off error accumulation, but I'm not convinced of that. (Actually, it probably is due to a zero/zero condition)
I wanted to either prove or disprove this identity, but I've been too busy and can't justify doing this as part of my work since it's just a point of curiousity.
I thought someone here might enjoy tackling this one. I suspect the identity is true, and there is probably a simple elegant way to prove it. I thought of doing out a Taylor expansion, but that looks tedious.
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