Fractional calculus trig (1 Viewer)

[Sien]

How do you do
Sinx/2x
And
3x+4/sin5x
Thanks


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[turntaker]

How do you do
Sinx/2x
And
3x+4/sin5x
Thanks


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What do you mean...

 

[turntaker]

Integrals?

 

[Sien]

Integrals?

Differentiate lol


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[turntaker]

Differentiate lol


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Quotient rule m8

 

[Sien]

Quotient rule m8

O shit I totally forgot about quotient rule thanks lel


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[turntaker]

O shit I totally forgot about quotient rule thanks lel


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GG no re

 

[seanieg89]

By the way, you should probably avoid the phrase "fractional calculus" in future.

This phrase already has a very specific meaning (involving things like taking "half" of a derivative of a function).

 

[braintic]

By the way, you should probably avoid the phrase "fractional calculus" in future.

This phrase already has a very specific meaning (involving things like taking "half" of a derivative of a function).

What exactly does taking half a derivative mean (conceptually)?

 

[seanieg89]

What exactly does taking half a derivative mean (conceptually)?

I can't think of any simple concept to directly relate it to as you can with the operator and the concept of change. (You don't often get such a natural conceptualisation of the more advanced mathematical constructions.)

But to motivate it, it is pretty natural to work with polynomials in . This is precisely what constitutes a constant coefficient linear ODE, and these pop up EVERYWHERE.

This begs the question: If we can sensibly apply a polynomial to D (a linear operator), can we find a sensible and useful way to apply more general functions to D? Or other linear operators? These questions lead to the study of functional calculus, which is immensely useful in PDE, spectral theory, etc.

The operator of "half-differentiation" is just the square root of D. It obeys the properties that you would expect. For example, applying it twice to a given differentiable function will just give you the functions derivative.

 

[braintic]

Not sure why, but I cannot see what I assume are latex equations in your response.

Is it meaningful to talk about irrational derivatives? Or ..... complex derivatives?

 

[seanieg89]

Not sure why, but I cannot see what I assume are latex equations in your response.

Is it meaningful to talk about irrational derivatives? Or ..... complex derivatives?

Latex appears to be broken on this site right now, if you click the reply with quote button you will see the code for my simple notation (I just define D as the single variable differentiation operator).

And yes it is meaningful to talk about irrational derivatives (equivalently irrational powers of the operator D), as well as complex order derivatives. Even things like e^D or sin(D).

As with taking powers of real numbers though, we should not expect uniqueness, and choosing a domain for these things might be subtle.

Something kind of interesting about the non-integer powers of D are that these operators are no longer local. That is, Df(x) is no longer determined only by the behaviour of f near x. The geometry of your domain also plays a role, which is why there are lots of different ways of defining this operator exponentiation, which have slightly varying scopes.