Defining functions in an interesting way?

[Gerenuk]

What do you think is some interesting or/and sensible way to define functions like exp(), cos() provided basic algebra rules (including integration) are known?

I make a suggestion I came up with
$$\ln x:=\int_1^x \frac{\mathrm{d}t}{t}$$
from where key properties follow directly.
For example
$$\ln (xy)=\int_1^{xy} \frac{\mathrm{d}t}{t}=\int_{1/x}^y \frac{\mathrm{d}t}{t}=\int_1^y \frac{\mathrm{d}t}{t}+\int_{1/x}^1 \frac{\mathrm{d}t}{t}=\int_1^y \frac{\mathrm{d}t}{t}+\int_1^x \frac{\mathrm{d}t}{t}=\ln x+\ln y$$

If we define the inverse function to be
$$\exp:=\ln^{-1}$$
then from the logarithm rule
$$\exp(\ln(xy))=\exp(\ln x+\ln y)$$
$$\exp(\ln(\exp(a)\exp(b)))=\exp(\ln \exp a+\ln \exp b)$$
$$\exp(a)\exp(b)=\exp(a+b)$$

Also it follows easily that for $t=\exp x$
$$\frac{\mathrm{d}}{\mathrm{d}x}\exp(x)=\frac{1}{\frac{\mathrm{d}}{\mathrm{d}t}\ln t}=t=\exp x$$

And hence
$$\exp x=1+\int_0^x \exp t\mathrm{d}t=1+\int_0^x \left(1+\int_0^t \exp t'\mathrm{d}t'\right)\mathrm{d} t=1+x+\int_0^x \int_0^t \exp t'\mathrm{d}t'\mathrm{d} t=1+x+\frac{x^2}{2!}+\dotsb$$

And as I mentioned in another post I strongly support
$$\cos x:=\Re(\exp \mathrm{i}x)$$
$$\sin x:=\Im(\exp \mathrm{i}x)$$

So much for playing around with functions late at night

 

[Tac-Tics]

Of all these, only your first line of the natural log is a good definition. In fact, the last two aren't even true! (cos x = Real(exp(ix))... same with sine, you missed the i in there).

I say that, aside from the first (and I guess also exp = ln^-1...), none of these are definitions. They are all equations involving interesting functions, but an equation doesn't always determine a function.

I'd also argue that integration isn't an algebraic technique =-)

 

[Gerenuk]

Of all these, only your first line of the natural log is a good definition. In fact, the last two aren't even true! (cos x = Real(exp(ix))... same with sine, you missed the i in there).

Well, that's actually the whole point here. The first line is my definition and all others follow in a one-line prove from that. I include the imaginary "i" in a sec.

I'd also argue that integration isn't an algebraic technique =-)

No lawyer please:tongue2:
I could try to restate everything more correctly, but it's already understandable now.

 

[Office_Shredder]

What's wrong with defining them as power series?

 

[Gerenuk]

What's wrong with defining them as power series?

I think it would be harder to derive all the results above.

Also a power series seems to have a "more complicating" structure with all it's coefficient, than the above integral. I mean it's comparable, but also contains a nasty factorial.

 

[Office_Shredder]

Well, exp(ix) is meaningless since ix isn't in the range of ln(x), unless you're going to use the power series definition of exp anyway, in which case you might as well have just started with the power series, shown the derivative of exp is exp (which is fairly easy) and then gotten that ln was the inverse of exp in order to prove those properties anyway. I'm not really sure how you've gained anything by swapping around the definition

 

[Gerenuk]

Well, exp(ix) is meaningless since ix isn't in the range of ln(x), unless you're going to use the power series definition of exp anyway, in which case you might as well have just started with the power series, shown the derivative of exp is exp (which is fairly easy) and then gotten that ln was the inverse of exp in order to prove those properties anyway. I'm not really sure how you've gained anything by swapping around the definition

Please be constructive instead of conservative here
I wrote all the results and that includes the addition theorem. I can check which parts are easy to deduce for myself.

I haven't thought about the complex domain and I do know that for all practical purposes power series are used as definition.

But we are trying to be creative here, which means contributing new ideas instead of destroying half-baked ones. If one finds some new "natural" way, then one might get new results. If you stick to what everyone else does, you only get what everyone else already has.

So new ideas welcome here!

 

[Hurkyl]

I think it would be harder to derive all the results above.

But, of course, it would be much easier to derive the power series for these functions.

 

[Hurkyl]

I would like to point out that no definition is going to be "easier" than any others -- whatever definition you choose, you still have to prove all of the same theorems, and the actual content of these proofs need not change.

For example, you find it easier to prove the algebraic properties of the logarithm from the integral formulation. Okay fine -- if you were to adopt a power series definition for the logarithm, you would first prove that the power series formulation implies the integral formulation (because you have to do that anyways), and then use the integral formulation to prove the algebraic properties.

 

[Office_Shredder]

But we are trying to be creative here, which means contributing new ideas instead of destroying half-baked ones. If one finds some new "natural" way, then one might get new results. If you stick to what everyone else does, you only get what everyone else already has.

What's your new idea though? You basically said we have two things that are equivalent, and you're going to change which one is the definition of exp. That doesn't seem like a game-changer to me... feel free to do it if you want, but by definition you're not going to get anything new out of it.

 

[Gerenuk]

But, of course, it would be much easier to derive the power series for these functions.

I find my derivation not mathematically rigorous, but a very simple one step process.

Okay fine -- if you were to adopt a power series definition for the logarithm, you would first prove that the power series formulation implies the integral formulation

That's the first reasonable comment here. :
Hmm, is it possible to prove the integral from the power series in an easy way?

What's your new idea though? ... feel free to do it if you want, but by definition you're not going to get anything new out of it.

Reread my post and also Hurkyls post. And please stop complaining, questioning the question and having a go at the ideas.

I repeat: This thread is asking about new ideas. My suggestion was a proposal, but not the topic or something that need consideration.

 

[Gerenuk]

I wanted to add that starting with my definition pi is defined by the variable "a" that satisfies
$$2a\mathrm{i}=\oint \frac{\mathrm{d}x}{x}$$
with the contour once around the origin.