I went trekking (to the Annapurna Base Camp this time) and, hence, left the math and physics books alone for a week or two. When I came back, it was like I had forgotten everything, and I wasn’t able to re-do the exercises. Back to the basics of complex numbers once again. Let’s start with Euler’s formula:
eix = cos(x) + isin(x)
In his Lectures on Physics, Richard Feynman calls this equation ‘one of the most remarkable, almost astounding, formulas in all of mathematics’, so it’s probably no wonder I find it intriguing and, indeed, difficult to grasp. Let’s look at it. So we’ve got the real (but irrational) number e in it. That’s a fascinating number in itself because it pops up in different mathematical expressions which, at first sight, have nothing in common with each other. For example, e can be defined as the sum of the infinite series e = 1/0! + 1/2! + + 1/3! + 1/4! + … etcetera (n! stands for the factorial of n in this formula), but one can also define it as that unique positive real number for which d(et)/dt = et (in other words, as the base of an exponential function which is its own derivative). And, last but not least, there are also some expressions involving limits which can be used to define e. Where to start? More importantly, what’s the relation between all these expressions and Euler’s formula?
First, we should note that eix is not just any number: it is a complex number – as opposed to the more simple ex expression, which denotes the real exponential function (as opposed to the complex exponential function ez). Moreover, we should note that eix is a complex number on the unit circle. So, using polar coordinates, we should say that eix is a complex number with modulus 1 (the modulus is the absolute value of the complex number (i.e. the distance from 0 to the point we are looking at) or, alternatively, we could say it is the magnitude of the vector defined by the point we are looking at) and argument x (the argument is the angle (expressed in radians) between the positive real axis and the line from 0 to the point we are looking at).
Now, it is self-evident that cos(x) + isin(x) represents exactly the same: a point on the unit circle defined by the angle x. But so that doesn’t prove Euler’s formula: it only illustrates it. So let’s go to one or the other proof of the formula to try to understand it somewhat better. I’ll refer to Wikipedia for proving Euler’s formula in extenso but let me just summarize it. The Wikipedia article (as I looked at it today) gives three proofs.
The first proof uses the power series expansion (yes, the Taylor/Maclaurin series indeed – more about that later) for the exponential function: eix = 1 + ix + (ix)2/2! + (ix)3/3! +… etcetera. We then substitute using i2 = -1, i3 = –i etcetera and so, when we then re-arrange the terms, we find the Maclaurin series for the cos(x) and sin(x) functions indeed. I will come back to these power series in another post.
The second proof uses one of the limit definitions for ex but applies it to the complex exponential function. Indeed, one can write ez (with z = x+iy) as ez = lim(1 + z/n)n for n going to infinity. The proof substitutes ix for z and then calculates the limit for very large (or infinite) n indeed. This proof is less obvious than it seems because we are dealing with power series here and so one has to take into account issues of convergence and all that.
The third proof also looks complicated but, in fact, is probably the most intuitive of the three proofs given because it uses the derivative definition of e. To be more precise, it takes the derivative of both sides of Euler’s formula using the polar coordinates expression for complex numbers. Indeed, eix is a complex number and, hence, can be written as some number z = r(cosθ+ isinθ), and so the question to solve here is: what’s r and θ? We need to write these two values as a function of x. How do we do that? Well… If we take the derivative of both sides, we get d(eix)/dx = ieix = (cosθ + isinθ)dr/dx + r[d(cosθ + isinθ)/dθ]dθ/dx. That’s just the chain rule for derivatives of course. Now, writing it all out and equating the real and imaginary parts on both sides of the expression yields following: dr/dx = 0 and dθ/dx = 1. In addition, we must have that, for x = 0, ei0 = [ei]0 = 1, so we have r(0) = 1 (the modulus of the complex number (1,0) is one) and θ(0) = 0 (the argument of (1,0) is zero). It follows that the functions r and θ are equal to r = 1 and θ = x, which proves the formula.
While these proofs are (relatively) easy to understand, the formula remains weird, as evidenced also from its special cases, like ei0 = ei2π = 1 = – eiπ = – e–iπ or, equivalently, eiπ + 1 = 0, which is a formula which combines the five most basic quantities in mathematics: 0, 1, i, e and π. It is an amazing formula because we have two irrational numbers here, e and π, which have definitions which do not refer to each other at all (last time I checked, π was still being defined as the simple ratio of a circle’s circumference to its diameter, while the various definitions of e have nothing to do with circles), and so we combine these two seemingly unrelated numbers, also inserting the imaginary unit i (using iπ as an exponent for e) and we get minus 1 as a result (eiπ = – 1). Amazing indeed, isn’t it?
[…] Well… I’d say at least as amazing as the Taylor or Maclaurin expansion of a function – but I’ll save my thoughts on these for another post (even if I am using the results of these expansions in this post). In my view, what Euler’s formula shows is the amazing power of mathematical notation really – and the creativity behind. Indeed, let’s look at what we’re doing with complex numbers: we start from one or two definitions only and suddenly all kinds of wonderful stuff starts popping up. It goes more or less like this really:
We start off with these familiar x and y coordinates of points in a plane. Now we call the x-axis the real axis and then, just to distinguish them from the real numbers, we call the numbers on the y-axis imaginary numbers. Again, it is just to distinguish them from the real numbers because, in fact, imaginary numbers are not imaginary at all: they are as real as the real numbers – or perhaps we should say that the real numbers are as imaginary as the imaginary numbers because, when everything is said and done, the real numbers are mental constructs as well, aren’t they? Imaginary numbers just happen to lie on another line, perpendicular to our so-called real line, and so that’s why we add a little symbol i (the so-called imaginary unit) when we write them down. So we write 1i (or i tout court), 2i, 3i etcetera, or i/2 or whatever (it doesn’t matter if we write i before the real number or after – as long as we’re consistent).
Then we combine these two numbers – the real and imaginary numbers – to form a so-called complex number, which is nothing but a point (x, y) in this Cartesian plane. Indeed, while complex numbers are somewhat more complex than the numbers we’re used to in daily life, they are not out of this world I’d say: they’re just points in space, and so we can also represent them as vectors (‘arrows’) from the origin to (x, y).
But so this is what we are doing really: we combine the real and imaginary numbers by using the very familiar plus (+) sign, so we write z = x + iy. Now that is actually where the magic starts: we are not adding the same things here, like we would do when we are counting apples or so, or when we are adding integers or rational or real numbers in general. No, we are adding here two different things here – real and imaginary numbers – which, in fact, we cannot really add. Indeed, your mommy told you that you cannot compare apples with oranges, didn’t she? Well… That’s exactly what we do here really, and so we will keep these real and imaginary numbers separate in our calculations indeed: we will add the real parts of complex numbers with each other only, and the imaginary parts of them also with each other only.
Addition is quite straightforward: we just add the two vectors. Multiplication is somewhat more tricky but (geometrically) easy to interpret as well: the product of two complex numbers is a vector with a length which is equal to the sum of the lengths of the two vectors we are multiplying (i.e. the two complex numbers which make up the product) , and its angle with the real axis is the sum of the angles of the two original vectors. From this definition, many things follow, all equally amazing indeed, but one of these amazing facts is that i2 = -1, i3 = –i, i4 = 1, i5 = i, etcetera. Indeed: multiplying a complex number z = x + iy = (x, y) with the imaginary unit i amounts to rotating it 90° (counterclockwise) about the origin. So we are not defining i2 as being equal to minus 1 (many textbooks treat this equality as a definition indeed): it just comes as a fact which we can derive from the earlier definition of a complex product. Sweet, isn’t it?
So we have addition and multiplication now. We want to do much more of course. After defining addition and multiplication, we want to do complex powers, and so it’s here that this business with e pops up.
We first need to remind ourselves of the simple fact that the number e is just a real number: it’s equal to 2.718281828459045235360287471 etcetera. We have to write ‘etcetera’ because e is an irrational number, which – whatever the term ‘irrational’ may suggest in everyday language – simply means that e is not a fraction of any integer numbers (so irrational means ‘not rational’). e is also a transcendental number – a word which suggest all kinds of mystical properties but which, in mathematics, only means we cannot write it as a root of some polynomial (a polynomial with rational coefficients that is). So it’s a weird number. That being said, it is also the so-called ‘natural’ base for the exponential function. Huh? Why would mathematicians take such a strange number as a so-called ‘natural’ base? They must be irrational, no? Well… No. If we take e as the base for the exponential function ex (so that’s just this real (but irrational) number e to the power x, with x being the variable running along the x-axis: hence, we have a function here which takes a value from the set of real numbers and which yields some other real number), then we have a function here which is its own derivative: d(ex)/dx = ex. It is also the natural base for the logarithmic function and, as mentioned above, it kind of ‘pops up’ – quite ‘naturally’ indeed I’d say – in many other expressions, such as compound interest calculations for example or the general exponential function ax = ex lna. In other words, we need this and exp(x) and ln(x) functions to define powers of real numbers in general. So that’s why mathematicians call it ‘natural’.
While the example of compound interest calculations does not sound very exciting, all these formulas with e and exponential functions and what have you did inspire all these 18th century mathematicians – like Euler – who were in search of a logical definition of complex powers.
Let’s state the problem once again: we can do addition and multiplication of complex numbers but so the question is how to do complex powers. When trying to figure that one out, Euler obviously wanted to preserve the usual properties of powers, like axay = ax+y and, effectively, this property of the so-called ‘natural’ exponential function that d(ex)/dx = ex. In other words, we also want the complex exponential function to be its own derivative so d(ez)/dz should give us ez once again.
Now, while Euler was thinking of that (and of many other things too of course), he was well aware of the fact that you can expand ex into that power series which I mentioned above: ex = 1/0! + x/1! + (x)2/2! + (x)3/3! +… etcetera. So Euler just sat down, substituted the real number x with the imaginary number ix and looked at it: eix = 1 + ix + (ix)2/2! + (ix)3/3! +… etcetera. Now lo and behold! Taking into account that i2 = -1, i3 = –i, i4 = 1, i5 = i, etcetera, we can put that in and re-arrange the terms indeed and so Euler found that this equation becomes eix = (1 – x2/2! + x4/4! – -x6/6! +…) + i(x – x3/3! + x5/5! -… ). Now these two terms do correspond to the Maclaurin series for the cosine and sine function respectively, so there he had it: eix = cos(x) + isin(x). His formula: Euler’s formula!
From there, there was only one more step to take, and that was to write ez = ex+iy as exeiy, and so there we have our definition of a complex power: it is a product of two factors – ex and eiy – both of which we have effectively defined now. Note that the ex factor is just a real number, even if we write it as ex: it acts as a sort of scaling factor for eiy which, you will remember (as we pointed it out above already), is a point on the unit circle. More generally, it can be shown that ex is the absolute value of ez (or the modulus or length or magnitude of the vector – whatever term you prefer: they all refer to the same), while y is the argument of the complex number ez (i.e. the angle of the vector ez with the real axis). [And, yes, for those who would still harbor some doubts here: ez is just another complex number and, hence, a two-dimensional vector, i.e. just a point in the Cartesian plane, so we have a function which goes from the set of complex numbers here (it takes z as input) and which yields another complex number.]
Of course, you will note that we don’t have something like zw here, i.e. a complex base (i.e. z) with a complex exponent (i.e. w), or even a formula for complex powers of real numbers in general, i.e. a formula for aw with a any real number (so not only e but any real number indeed) and w a complex exponent. However, that’s a problem which can be solved easily through writing z and w in their so-called polar form, so we write z as z = ¦z¦eiθ = ¦z¦(cosθ + isinθ) and w as ¦w¦ eiσ = ¦w¦(cosσ + isinσ) and then we can take it further from there. [Note that ¦z¦ and ¦w¦represent the modulus (i.e. the length) of z and w respectively, and the angles θ and σ are obviously the arguments of the same z and w respectively.] Of course, if z is a real number (so if y = 0), then the angle θ will obviously be zero (i.e. the angle of the real axis with itself) and so z will be equal to a real number (i.e. its real part only, as its imaginary part is zero) and then we are back to the case of a real base and a complex exponent. In other words, that covers the aw case.
[…] Wel… Easily? OK. I am simplifying a bit here – as I need to keep the length of this post manageable – but, in fact, it actually really is a matter of using these common properties of powers (such as ea+biec = e(a+c)+bi and it actually does all work out. And all of this magic did actually start with simply ‘adding’ the so-called ‘real’ numbers x on the x-axis with the so-called ‘imaginary’ numbers on the y-axis. 🙂
Penrose’s Road to Reality dedicates a whole chapter to complex exponentiation (Chapter 5). However, the development is not all that simple and straightforward indeed. The first step in the process is to take integer powers – and integer roots – of complex numbers, so that’s zn for n = 0, ±1, ±2, ±3… etcetera (or z1/2, z1/3, z1/4 if we’re talking integer roots). That’s easy because it can be solved through using the old formula of Abraham de Moivre: (cosθ + sinθ)n = cos(nθ) + isin(nθ) (de Moivre penned this down in 1707 already, more than 40 years before Euler looked at the matter). However, going from there to full-blown complex powers is, unfortunately, not so straightforward, as it involves a bit of a detour: we need to work with the inverse of the (complex) exponential function ez, i.e. the (complex) natural logarithm.
Now that is less easy than it sounds. Indeed, while the definition of a complex logarithm is as straightforward as the definition of real logarithms (lnz is a function for which elnz = z), the function itself is a bit more… well… complex I should say. For starters, it is a multiple-valued function: if we write the solution w = lnz as w = u+iv, then it is obvious that ew will be equal to eu+iv = eueiv and this complex number ew can then be written in its polar form ew = reiθ with r = eu and v = θ + 2nπ. Of course, ln(eu+iv) = u + iv and so the solution of w will look like w = lnr + i(θ + 2nπ) with n = 0, ±1, ±2, ±3 etcetera. In short, we have an infinite number of solutions for w (one for every n we choose) and so we have this problem of multiple-valuedness indeed. We will not dwell on this here (at least not in this post) but simply note that this problem is linked to the properties of the complex exponential function ez itself. Indeed, the complex exponential function ez has very different properties than the real exponential function ex. First, we should note that, unlike ex (which, as we know goes from zero at the far end of the negative side of the real axis to infinity as x goes big on the positive side), ez is a periodic function – so it oscillates and yields the same values after some time – with this ‘after some time’ being the periodicity of the function. Indeed, ez = ez +2πi and so its period 2πi (note that this period is an imaginary number – but so it’s a ‘real’ period, if you know what I mean :-)). In addition, and this is also very much unlike the real exponential function ex, ez can be negative (as well as assume all kinds of other complex values). For example, eiπ = -1, as we noted above already.
That being said, the problem of multiple-valuedness can be solved through the definition of a principal value of lnz and that, then, leads us to what we want here: a consistent definition of a complex power of a complex base (or the definition of a true complex exponential (and logarithmic) function in other words). To those who would want to see the details of this (i.e. my imaginary readers :-)), I would say that Penrose’s treatment of the matter in the above-mentioned Chapter 5 of The Road to Reality is rather cryptic – presumably because he has to keep his book around 1000 pages only (not a lot to explain all of the Laws of the Universe) and, hence, Brown & Churchill’s course (or whatever other course dealing with complex analysis) probably makes for easier reading.
[As for the problem of multiple-valuedness, we should probably also note the following: when taking the nth root of a complex number (i.e. z1/n with n = 2, 3, etcetera), we also obtain a set of n values ck (with k = 0, 1, 2,… n-1), rather than one value only. However, once we have one of these values, we have all of them as we can write these ck as ck = r1/nei(θ/n+2kπ/n), (with the original complex number z equal to z = reiθ) then so we could also just consider the principal value c0 and, as such, consider the function as a single-valued one. In short, the problem of multiple-valued functions pops up almost everywhere in the complex space, but it is not an issue really. In fact, we encounter the problem of multiple-valuedness as soon as we extend the exponential function in the space of the real numbers and also allow rational and real exponents, instead of positive integers only. For example, 41/2 is equal to ±2, so we have two results here too and, hence, multiple values. Another example would be the 4th root of 16: we have four 4th roots of 16: +2, -2 and then two complex roots +2i and -2i. However, standard practice is that we only take the positive value into account in order to ensure a ‘well-behaved’ exponential function. Indeed, the standard definition of a real exponential function is bx = (elnb)x = elnbex, and so, if x = 1/n, we’ll only assign the positive 4th root to ex. Standard practice will also restrict the value of b to a positive real number (b > 0). These conventions not only ensures a positive result but also continuity of the function and, hence, the existence of a derivative which we can then use to do other things. By the way, the definition also shows – once again – why e is such a nice (or ‘natural’) number: we can use it to calculate the value for any exponential function (for any real base b > 0). But so we had mentioned that already, and it’s now really time to stop writing. I think the point is clear.]