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Edit : This question is about not about proving identities, but representations that are easier to work with than Taylor series or integral definition for the functions exp or ln functions. Please do not attempt to prove identities, just want alternative initial representations. For example the infinite zeros of sin are not obvious with the Taylor series definitions, where as Euler Product formula $\sin(x) = x\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right)$ makes it trivial to see sin,cos have infinite many zeros on the real line.

Are there better alternative definitions than $\exp(x) = {\large\sum\limits_{k=0}^\infty} \dfrac{x^k}{k!} , \ln(x) = {\large\int_1^x} \dfrac1t\ dt$. that can be used for derivation of their identities e.g. $\exp (x+y)=\exp(x)\exp(y)$, $\ln (xy) =\ln (x)\ln(y)$ (and by extension sin and cos related) identities?

Is there any other starting point other than $\exp(x) = {\large\sum\limits_{k=0}^\infty} \dfrac{x^k}{k!}$ , where this definition of exponential function is obtained as one of the possible representations of exponential function? If yes, what body of theory deals with these type of questions (if any).

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Well, one way to see the exponential is to realize the following:

There are two canonical group structures in $\mathbb{R}$: $(\mathbb{R},+)$ and $(\mathbb{R}_{>0}, \cdot)$. The exponential is, together with the identity map, the only two "interesting" functions*: the intersection of algebra and analysis - The identity map is the unique continuous isomorphism from the additive structure to itself that makes $f(1)=1$ and the exponential map is the unique continuous isomorphism from the additive structure to the multiplicative structure that makes $f(1)=e$ (the $e$ here just enters as a suitable distinguishing point). This is a good way to think about things, but as I said in the comments, is not nearly as workable as the series definition. And I think no other definition will be. Power series are simple, well-understood, highly computational etc. (See below also for the "differential equation definition")

*-In fact, all other elementary functions can be obtained by them: polynomials, $\sin$, $\cos$, $\log$ etc.


This question admits a lot of different answers. I'll approach one of the problems you mentioned: proving $\exp(x+y)=\exp(x)\exp(y)$. One can arrive at this from the Cauchy product formula, but also from the simple fact that $\exp'(x)=\exp(x)$ and $\exp(0)=1$ (which is something that follows easily from the power series representation, and can also be taken as the definition of the exponential, that is, the function that solves the differential equation $y'=y$ with initial condition $y(0)=1$).

For that, define, for fixed $y \in \mathbb{R}$, $$f(x)=\frac{\exp(x+y)}{\exp(x)\exp(y)}.$$

We then have $$f'(x)=\frac{\exp(x)\exp(y) \exp(x+y)-\exp(x+y)\exp(x)\exp(y)}{\exp(x)^2\exp(y)^2}=0,$$ by the quotient rule. Therefore, $f(x)$ is constant. Since it is clear that $f(0)=1$, we have that $\exp(x+y)=\exp(x)\exp(y)$ for all $x$. But $y$ is also arbitrary, hence we have the equality for all $x,y \in \mathbb{R}$.

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  • $\begingroup$ Apologies, was not after a proof of identities, but something other than Taylor series or integral definition for the functions $\endgroup$ – Arjang Dec 14 '15 at 5:04
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    $\begingroup$ @Arjang As I mentioned, you can take as definition that $e^x$ is the function which solves the differential equation $y'=y$ with $y(0)=1$. Maybe this is more intuitive, but not very workable. The series definition for $\exp(x)$ is arguably the most workable definition one has. From that, you can define $\log$ to be its inverse function, and its integral form is a quick result from the chain rule and Fundamental Theorem of Calculus. $\endgroup$ – Aloizio Macedo Dec 14 '15 at 5:06
  • $\begingroup$ Is there a more general theory that makes the identities obvious, like Euler Product folmula for sin,cos shows the infinite zeros on the real line by definition, the series definition of sin,cos are a pain to use for that. $\endgroup$ – Arjang Dec 14 '15 at 5:10
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I'm pretty partial to using Hyperbolic Functions, so here's one mildly obtuse way to manipulate alternative defintions of the exponential function.

Since $\sinh(x) = \dfrac{e^x - e^{-x}}{2}$ and $\cosh(x) = \dfrac{e^x + e^{-x}}{2}$ (which, admittedly, is a bit circular for your question's sake...but whatever), it is easy to show that $e^{\pm x} = \cosh(x) \pm \sinh(x)$.

So what's the point? Well, you also probably have to know the analogy to the great-grand-daddy of all trig identities with $\cos^2(x) + \sin^2(x) = 1$:

$$\cosh^2(x) - \sinh^2(x) = 1.$$

Okay, now that you've got that, you can go right into manipulating expressions in a very, very similar way to the way you would prove identities involving sines and cosines.

I don't feel like I've answered the question in the scope that you request, but I hope this provides you with a different perspective on using alternative defintions to the exponential.

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When I learned the definitions of these functions, we defined $\ln(x) = {\large\int_1^x} \dfrac1t\ dt$, from which we were able to derive the property $ln(ab)=ln(a)+ln(b)$ as follows $$ln(ab)=\ln(x) = {\int_1^{ab}} \dfrac1t\ dt= \int_1^{a} \dfrac1t\ dt +\int_a^{ab} \dfrac1t\ dt $$

letting $t=ax$ $$\int_1^{a} \dfrac1t\ dt +\int_1^{b} \dfrac1{ax}\ d(ax)=ln(a)+ln(b)$$

Then, you can define $e^x$ as the inverse of $ln(x)$ and take the natural log of $e^ae^b$ $$ln(e^ae^b)=ln(e^a)+ln(e^b)=a+b$$ now undo the natural log using the exponential and you get $e^{a}e^b=e^{a+b}$ This gives a helpful primer on the exponential before continuing on to the hyperbolic functions in Xoque55's answer

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