Way ahead for Trig Integral

[sethoflagos]

The physics group have redirected me here.

 

All parameters are constant save the Delta t terms.

 

 

Related but a bit more involved is

 

The divisor is essentially a large positive value with a small 'wiggle', but it would be nice to have an analytic solution.

 

Many thanks for your time.

 

Seth

[blue89]

do not miss the time there ...

you may make your such questions be solved only with some programmes like WOLFRAM ...

this programme will solve highly quickly and ...this will be the best way ..be sure.

NOTE: I don't mean I cannot solve such problems or this for any other ones.

but I would say that this will not be logical. (if you were asking something theoritic ,it would not be so)

you may see so interestingly difficult or more difficult quations in Nature's or SCIENCE's articles ,but I do not think that those scientists who prepared that articles had calculated all that integral fomrs by their hand ..

wolfram is expertise to calculate such problema (I do not know using this well ,but I saw this at once when I was at BSc in one course)

wolfram is good.

Edited August 9, 2016 by blue89

[sethoflagos]

Unfortunately, I do not know using this thing well either, blue89.

 

I was brought up with log tables and slide rules.

 

Wolfram is a mystery to me.

[Keen]

If you do not want to use Wolfram, you're going to have a difficult time calculating this integral. The general method for calculating these kinds of integrals is using following formulas: [math]sin(x)=\frac{2tan(\frac{x}{2})}{1+tan(\frac{x}{2})^2}[/math] and [math]cos(x)=\frac{1-tan(\frac{x}{2})^2}{1+tan(\frac{x}{2})^2} [/math]

You then change the integration variable by putting [math]u=tan(\frac{x}{2})[/math] and you should obtain a rational function in u. You then decompose it into partial fractions and integrate these.

 

However considering how many different constants there are, it is likely to take enormous time and I definitely am not motivated to do this kind of calculations.

[sethoflagos]

Many thanks!

 

I did eventually get some joy out of Wolfram, and the results actually simplify very nicely (or at least better than expected.)

 

But it's good to have a longhand approach available for checking purposes.

 

Seth

[sethoflagos]

Over the last few hours, I've been experimenting with the Wolfram free calculator.

I hope you guys appreciate what a tremendous advantage this facility provides compared to what previous generations had access to.

It solves in seconds stuff that used to take me weeks to work through by hand.

But you know that don't you?

[Sensei]

Over the last few hours, I've been experimenting with the Wolfram free calculator.

 

I hope you guys appreciate what a tremendous advantage this facility provides compared to what previous generations had access to.

 

It solves in seconds stuff that used to take me weeks to work through by hand.

 

But you know that don't you?

 

That's what for are computers.. To compute

 

If some application has not sufficient built-in features (or they're too slow), you can always make your own application (with little GUI or command-line).

You will spend time writing program, and get result quickly,

instead of spending time calculating everything on paper.

 

From time to time, it's good to make some computing manually, though.

[sethoflagos]

 

That's what for are computers.. To compute

 

If some application has not sufficient built-in features (or they're too slow), you can always make your own application (with little GUI or command-line).

You will spend time writing program, and get result quickly,

instead of spending time calculating everything on paper.

 

From time to time, it's good to make some computing manually, though.

 

I'm quite familiar with solving systems such as the one I'm working on now, by numerical methods on computer (since the late '70s). That's bread and butter for some of the work I've done in the past, and indeed it is how I would normally have approached the system I'm currently working on. Pretty standard for hyperbolic PDEs. Clients pay for results in a cost-effective time-frame, not the beauty of the mathematics.

 

But to have fully analytic solutions readily available to really quite complex systems is pretty mindblowing!

Edited August 13, 2016 by sethoflagos

[sethoflagos]

Just to show that your advice has yielded concrete results, here's one of the pretty pictures that comes out of it.

 

 

Which indicates that a simple approximate model deviates from the fully analytic solution (that's taken me maybe a month to derive) by about 1 degree of phase shift here and there. May as well not have bothered!

 

Thanks again anyway!

Seth