[Mika]

Arrived in slightly different special function, maybe someone could explain why?
And how the hell you can put up those neato integrate signs here?

Integrate:
ln(tan x) dx  || Substituted tan(x)=u and solved dx = 1/(1+u^2) du

Arrived at:
INT { ln(u)*[1/(1+u^2)] }du, partial integration resulted in

ln(u)*arctan(u) - INT { arctan(u)/u }du, which results in a special function also.

The handbook of mathematical functions, (Abramowitz-Stegun) states that:
INT { arctan(u) / u } du = INT from 0 to INF { exp(-u*t)*si(t) }dt,
where si is a special function also, defined on the same book.

It was probably something like INT from Z to INF { (sin t) / t }dt.

Mika

[CP5670]

Arrived in slightly different special function, maybe someone could explain why?

Many of them are related to each other. There are three or four fundamental classes of functions (hypergeometric, zeta/Lerch and elliptic come to mind) and the others are all basically combinations or special cases of those. The si (sine integral) and polylog are both of the hypergeometric variety.

There are a lot of essentially redundant functions in use since they often came from different contexts. I know si for example is used in speech recognition, even though it's just a slight variant of the arguably more basic ei.

And how the hell you can put up those neato integrate signs here?

You can find it in charmap. Getting the superscripts and subscripts to line up can be annoying though. I think it's clearer to just write something like Integral[ log(tan(x)) , {x,0,pi}].

[Mongoose]

This is exactly why I despised integrals so much.  Derivatives are such beautiful, elegant things...learn a few simple rules, and you can derive out even the most complex function.  But as soon as you get into integration, you realize that the rules are largely useless, as there are gobs of types out there that are flat-out impossible to do.  This is about the point in my schooling when I decided to yell a very hearty "**** you!" to math.

[CP5670]

This is exactly why I despised integrals so much.  Derivatives are such beautiful, elegant things...learn a few simple rules, and you can derive out even the most complex function.  But as soon as you get into integration, you realize that the rules are largely useless, as there are gobs of types out there that are flat-out impossible to do.  This is about the point in my schooling when I decided to yell a very hearty "**** you!" to math.

The funny thing is that this is not true in general. It just happens that all the elementary functions (rational functions, exp and log, and combinations of those) have derivatives that can be easily written out.

The most basic function you encounter beyond that setting is the gamma function (factorial), and there you already run into a situation where you have to introduce a new function for every derivative.

[redsniper]

Oh, kind of like: Heaviside(x)'= delta(x) ?

[CP5670]

Yes, that's another example, although the derivative there is only meaningful in a weak/distributional sense.

[Kosh]

ln(tan x) dx  || Substituted tan(x)=u and solved dx = 1/(1+u^2) du

I should have figured it was something like that. Thanks.

[Mika]

No problem. Physicists know these type of integrals by heart.

Mika

[Mongoose]

No problem. Physicists know these type of integrals by heart.

Good physicists do.  I, unfortunately, do not.

And I suppose it's a good thing I got out of math when I did, CP.  I don't need another excuse to dislike its higher workings.

[Dark RevenantX]

****.  I'll have to do this in AP Calculus BC for my Senior year, won't I?



Though, I have done harder things...  Such as creating a single formula that calculates the exact horizontal speed that a projectile must have in order to hit a certain point, taking into effect the change in elevation, gravity, actual speed of the projectile, etc (not including air resistance and such).

[Mika]

Though, I have done harder things...  Such as creating a single formula that calculates the exact horizontal speed that a projectile must have in order to hit a certain point, taking into effect the change in elevation, gravity, actual speed of the projectile, etc (not including air resistance and such).

I don't think that is yet too bad. Try solving the equation of the projectile in rotating or accelerating coordinate system. Especially rotating coordinate system (like Earth) introduces several rather interesting forces into the play. Did it once, spherical coordinate system was a must on Earth's case. Also do try showing that the effect of the Coriolis force is neglible when the projetile travels 300 m for example...

Lagrange's formulation of mechanics helps a lot of these analysis. The only inherent problem in it is that it cannot handle friction.

Mika