Why is the exponential function the most important?

Question

In the prologue to his Real and Complex Analysis, Walter Rudin states matter-of-factly that $\exp$ is the most important function in mathematics (take note, not merely in analysis). This looked so incredible to me, and too much to claim (out of the infinite variety of conceivable mathematical functions extending beyond mapping subsets of $\mathbb{C}$ to other such subsets) that I began to wonder if this sentiment is a common one (I heard a math prof saying it's the second most important function -- he said this only as an afterthought; he had indeed claimed that it was the most important) and if so, what could so distinguish this function like this.

So, my question is twofold:

1. Do you think, or is it common knowledge to mathematicians, that the exponential function is indeed the most important in all of mathematics?

2. If so, why is this the case? What could make it more important than all the conceivable and inconceivable functions (linking arbitrary pairs of sets) out there?

Thank you.

Answer 1

To answer your second question:

Our entire civilization hinges on our ability to model and manipulate the surrounding environment.

The natural exponential function lies at the heart of this ability. It solves the prototypical ordinary differential equation $x' = \lambda x$ and its extension to matrices solves the system $x' = Ax$. This allows us to analyze and understand problems in classical mechanics, electromagnetism and acoustics.

In particular, without understanding the natural exponential function there would be no infrastructure, no mechanical transportation and no electricity.

More sophisticated mathematics has allowed us to develop quantum mechanics and cryptography. The utility of these topics is indisputable, but they are also irrelevant until a certain level of technology is attained. Solving Newton's laws of motion is useful at a far earlier stage of development.

To answer your first question:

No, I do not believe that this is common knowledge to all mathematicians. It is necessary to have knowledge of the history of science and technology to appreciate the importance of the natural exponential function.

Answer 2

I'll prepend the above question 1. and 2. by a question 0.,

$$ ----- $$ QUESTION 0.   In what sense can we talk about a mathematical notion, say a function $\,F,\,$ being the most important among all of them? $$ ----- $$

One may talk about the ubiquity of $\, F,\, $ or about its engineering-like logical necessity, etc.

A time ago I've asked this question at MO in the context of education. I asked about the crucial place/function which is a central point for learning mathematics. You aim at it, and then you go from there in all possible directions.

Frankly speaking, I asked about two stages. The first one was the field of complex numbers $\,\Bbb C.\,$ And I had also my own answer for the 2nd stage hiding, waiting for reactions from others. Sure enough, one (or more?) participants mentioned the complex exponential function -- of course complex since it had to follow the field of complex numbers.

My own answer was nearly equivalent, I consider that this most important function is the complex logarithm.

Now back to the OP's questions.

$$ ----- $$ QUESTION 1.   Do you think, or is it common knowledge to mathematicians, that the exponential function is indeed the most important in all of mathematics? $$ ----- $$

Certainly, some people would say that the zeta function is the most important. It encodes the secrets of prime numbers so well.

I would consider the partition function of the statistical physics at least as important as any other. It's the fundamental ingredient of the fundamental law of statistical mechanics. Richard Feynman said:

• ... the entire subject is either the slide-down from this summit ... or the climb-up ...*

(of course, the exponential function serves the partition function faithfully).

Thus, to answer Q1, it is NOT common knowledge, and for two reasons. (A) people simply don't know or don't care to think about such issues; (B) some mathematicians may have a different opinion, it all depends.

Nevertheless, if we talk about learning mathematics and having a wide open road for doing research than indeed the complex logarithm would be it.

 

$$ ----- $$ QUESTION 2.   If so, why is this the case? What could make it more important than all the conceivable and inconceivable functions (linking arbitrary pairs of sets) out there? $$ ----- $$

This question is really two questions: internal (atomic) and external (the world view). Namely: (A) What's so wonderful about the making of function $\,\log\,$ (about its structure/construction) that this construction is so helpful around mathematics? (B) What are the crucial achievements of function $\, \log\, $ and how deeply $\, \log\, $ is intertwined with nearly all mathematics?

(A)   The internal answer, from the inside of $\,\exp/\log,\,$ is that they connect tightly the two most important elementary operations, the complex addition, and multiplication -- this is a true miracle (without which mathematics wouldn't be the same).

(B)   First, we need to talk about the field of complex numbers $\,\Bbb C\,$ -- it provides the full scope for $\,\exp/\log.\ $ In particular, $\,C\,$ is truly Euclidean plane geometry. It's a great shame that children are not taught Euclidean Geometry via complex numbers, they would master the complex numbers and Euclidean Geometry by the end of the 5th great, no sweat (well, poor teachers, but that's a different story).

Now, $\ \exp/\log\ $ are the very first transcendental function, they are not algebraic (you can't obtain them by the five arithmetic operations), and they form a straight bridge between algebra and analysis; you can say -- between finiteness and infinity.

The logarithmic function is at the start of algebraic topology. That's where the true wonders of topology and its complexity shine. You may recall the Eilenberg's theorem about dissecting $\, Bbb\,$ which is a concert played on two instruments, $\, \exp\,$ and $\,\log.$

Of course, these two functions appear anywhere you look at, e.g. algebra, combinatorics, geometry, probability, statistical mechanics, ...

One could praise these two guys forever.

PS   Statistical Mechanics is so rich and complete that if mathematicians were restricted to Statistical Mechanics only, it'd make hardly any difference, they wouldn't miss much.

Answer 3

It is an interesting question I asked myself many times. There is to-be-finished list why the exponential function is important (if not mostly by all can agree. Welcome to correct and improve...):

1. It represents Hilbert basis which forms an orthonormal basis of the space of functions with finite Lebesgue integrals...
2. It has true "integrity" which means that she never hide how to change, i.e., the growth or decay is just herself for your observation as time goes by,
3. It embrase universe (all directions in space) so that you can scan by delta functions,
4. It builds a nice bridge to expand number system or the fields,
5. Functional representation (Fourier, Laplace, wave equations...)
6. ...