Complex numbers 2: Euler's formula, complex powers and logarithms

Complex numbers 2

Euler's formula and identity.
Complex powers and logarithms

1 - The complex number \( cos \; \alpha + i\; sin\; \alpha \)

The particular importance of this number in the representation of complex numbers leads us to review and look more closely at some of its important properties.
To keep the notation lighter, from now on we will write \( cos \; \alpha + i\; sin\; \alpha \) instead of the more correct expression \( cos(\alpha) + i\; sin(\alpha) \), and the same for all trigonometric functions, wherever this cannot cause ambiguity.

1.1 - More properties of \( cos \; \alpha + i\; sin\; \alpha \)

So let us go back to the complex number \( cos \; \alpha + i\; sin\; \alpha \).
As we saw in part 1, multiplied by its modulus \(r_z = |\dot{z}| \), it can represent any complex number: \( \dot{z}= r_z\; (cos \; \alpha + i\; sin\; \alpha) \).
But this object has other interesting aspects that are worth investigating.
For convenience, let us write \( \dot{u}_{\alpha} = cos \; \alpha + i\; sin\; \alpha \).
We use the letter \(u \), which reminds us of unity, because its modulus is indeed one: \( |\dot{u}_{\alpha}|^2 = cos^2 \; \alpha + sin^2 \; \alpha = 1 \), and the subscript \(\alpha \) to highlight the argument .
The argument of \(u_{\alpha} \), written \(Arg(u_{\alpha}) \), is the angle \(\alpha \), measured in radians. It can take all values between \(0 \) and \(2 k \pi \), with \(k = 1, 2, ... \) an integer. To remove the ambiguity due to the periodicity of the sine and cosine functions, it is often restricted to the interval \(-\pi\; \leq \;\alpha\; \leq\; \pi \). In this case it is written \(arg(u_{\alpha})\) and is called the principal value of the argument.
The complex conjugate of \(u_{\alpha} \) is clearly \( \dot{u}^*_{\alpha} = cos \; \alpha - i\; sin\; \alpha\), but the interesting thing is that it is the same as the reciprocal of \(u_{\alpha} \):
\[ \begin{split} \frac{1}{\dot{u}_{\alpha}} &= \frac{1}{\cos\alpha + i\sin\alpha} = \frac{\cos\alpha - i\sin\alpha}{(\cos\alpha + i\sin\alpha)(\cos\alpha - i\sin\alpha)} \\[1ex] &= \frac{\cos\alpha - i\sin\alpha}{\cos^2\alpha + \sin^2\alpha} = \cos\alpha - i\sin\alpha \end{split} \] So: \[\dot{u}^*_{\alpha} = \frac{1}{\dot{u}_{\alpha}} \] De Moivre's formula (section 4.3 - part 1): \[ (cos \; \alpha + i\; sin\; \alpha)^n = cos \;n \alpha + i\; sin\; n \alpha\] makes it easy to compute the n-th power of \(u_{\alpha} \).
Here a peculiar behavior of \(Arg(u_{\alpha}) \) shows up, made more evident by the polar form \(\dot{u}_{\alpha} = 1\; \angle\; \alpha \) :
\[(\dot{u}_{\alpha})^n = (1 \angle \alpha)^n = 1^n \angle n\alpha = 1 \angle \alpha+\alpha+\alpha+\dots = 1 \; \angle {\small\sum_{1}^n} \alpha \] that is: \[ 1 \;\angle \;\alpha \cdot 1 \;\angle \;\alpha \cdot 1 \;\angle \;\alpha\;\cdot ... = 1\; \angle\; \alpha + \alpha + \alpha+ ... \quad (n\; times)\]
In this context, \(Arg(u_{\alpha}) \) seems to behave somewhat like a logarithm, in the sense that, from a product of n quantities depending on \(\alpha \), we get a resulting quantity that depends on the sum of n terms \(\alpha \). This cannot help but remind us, in some way, of logarithms, which have precisely the property of turning products into sums ⁽¹⁾.
A link between \(\alpha = Arg(u_{\alpha}) \) and the logarithms of complex numbers does indeed exist, but it is not as simple and evident as in the case of real numbers. We will look into it later.

1.2 - The product \(\dot{u}_{\alpha}\,\dot{u}_{\beta}\)

De Moivre's formula solves the particular case of computing powers of \(\dot{u}_{\alpha} \), that is, the product of \(\dot{u}_{\alpha} \) by itself n times. In part 1 we showed that, in general, the product of two complex numbers is obtained by multiplying their moduli and adding their arguments: \[\dot{u}_{\alpha}\,\dot{u}_{\beta} = 1 \angle \;\alpha \cdot 1 \angle \;\beta = 1 \angle \; \alpha+\beta\] Now let us think of \(\alpha \) and \(\beta \) as two real variables and, to reinforce this idea, let us call them x and y. We will then write: \[ \dot{u}(x) = cos \; x + i\; sin\; x \quad \dot{u}(y) = cos \; y + i\; sin\; y \] highlighting the dependence on the arguments x and y, which of course represent angles. From what we have said and proved so far, we find that: \[\dot{u}(x)\,\dot{u}(y) = \dot{u}(x+y)\] Among the real functions of a real variable that we know, there is a function with a similar behavior: the exponential function \(a^x \) \[a^x\,a^y = a^{x+y}\] the question that naturally arises at this point is: "is there perhaps some connection between \( cos \; x + i\; sin\; x\) and the exponential function?"
Before answering, let us look at another important feature of \(\dot{u}_{\alpha} \).

1.3 - The derivative of \( cos \; x + i\; sin\; x\) with respect to its argument

To make things clearer, here too let us use the more familiar functional notation \( \dot{u}(x)\). Let us then compute the derivative \( \dot{u}'(x)\) with respect to x, applying the usual rules of differentiation:
\[ \dot{u}'(x) = \frac{d}{dx}(cos\; x + i\; sin\; x) = \frac{d\; cos\; x}{dx} + i\; \frac{d\; sin\;x}{dx} = - sin\; x + i \; cos\; x\] since \( i^2 = -1\), we have \( -sin x = i^2 sin x\), so, factoring out \(i\):
\[ \frac{d}{dx}(cos\; x + i\; sin\; x) = i^2\: sin\; x + i \; cos\; x = i\;(cos\;x + i\;sin\;x) \] and finally: \[ \dot{u}'(x) = i\;\dot{u}(x)\] So taking the derivative of \( \dot{u}\) with respect to its argument is equivalent to multiplying \( \dot{u}\) by the imaginary unit, which gives \( \dot{u}'= i\,u\).
From the vector point of view, we know that this means rotating \( \dot{u}\) by 90° counterclockwise. On the other hand, if we differentiate again, we get: \[ \dot{u}''(x) = \frac{d^2\;\dot{u}(x)}{dx^2} \ = \frac{d \dot{u}'(x)}{dx} = i\;\frac{d \dot{u}(x)}{dx} = i^2\;\dot{u}(x) \] and, in general: \[ \dot{u}^{(n)}(x) = \frac{d^n}{dx^n} \dot{u}(x) = i^n\;\dot{u}(x) \] the n-th order derivative of \( \dot{u}(x)\) is \( \dot{u}(x)\) multiplied by the n-th power of the imaginary unit.
This behavior is surprisingly similar to that of the exponential function \( e^{cx}\): \[ \frac{d^n\;e^{c x}}{dx^n} = c^n\;e^{c x} \quad c,\;x:\; real,\; n= 1, 2, 3,..\] If we could set \( c = i\), which means being able to define an exponential with an imaginary exponent, \( e^{i x}\), the correspondence would be complete.

2 - Euler's formula

2.1 - Deriving the Euler-Cotes formula

Looking at the properties of \( \dot{u}_\alpha = cos \; \alpha + i\; sin\; \alpha\), we have found several signs suggesting that there is a deep relationship between it and the exponential function. Let us sum up these clues:

1. the argument α tends to behave like a logarithm, and therefore like an exponent;
2. the product follows the typical rules of exponentials;
3. differentiation gives a result similar to the one obtained by differentiating the exponential \( e^{c x}\).

So let us see whether we can satisfactorily prove that this relationship exists.

In section 5.5 of part 1 we found the expression for the n-th roots of a complex number. In the particular case of \( \dot{u}_\alpha\):
\[\sqrt[n]{\dot u}_\alpha = \dot{u}_\alpha^{\frac{1}{n}} = cos \frac{ \alpha +2 k \pi}{n} + i\;sin \frac{ \alpha +2 k \pi}{n} \hspace{1cm} k = 0, 1, 2, ..., n-1 \] restricting ourselves to k = 0, we find \(\dot{u}_\alpha^{\frac{1}{n}} \) \[\dot{u}_\alpha^{\frac{1}{n}} = cos \frac{ \alpha}{n} + i\;sin \frac{ \alpha}{n} \] Raising it to the n-th power we get \( \dot{u}_\alpha\) back, since raising to the n-th power is the inverse operation of taking the n-th root: \[\left(\dot{u}_\alpha^{\frac{1}{n}} \right)^n = \left( cos \frac{ \alpha}{n} + i\;sin \frac{ \alpha}{n}\right)^n = cos \frac{ n\;\alpha}{n} + i\;sin \frac{ n\;\alpha}{n} = \dot{u}_\alpha \] So, for every integer n, it is always true that:
\[ \left( cos \frac{ \alpha}{n} + i\;sin \frac{ \alpha}{n}\right)^n = cos \;\alpha+ i\;sin \alpha \] Now imagine increasing n so that it takes larger and larger values.
The mathematical operation is taking the limit as n tends to infinity:

\[ \lim_{n \to \infty}{ \left( cos \frac{ \alpha}{n} + i\;sin \frac{ \alpha}{n}\right)^n} = cos \;\alpha+ i\;sin \alpha \] In this case \[ \lim_{n \to \infty}{\frac{ \alpha}{n}}= 0 \quad \Rightarrow \quad \left( \lim_{n \to \infty} {cos \frac{ \alpha}{n}}=1\; \wedge\; \lim_{n \to \infty} {sin \frac{ \alpha}{n}} = \frac{ \alpha}{n} \right) \] Then it becomes possible to write
\[ \lim_{n \to \infty}{ \left( cos \frac{ \alpha}{n} + i\;sin \frac{ \alpha}{n}\right)^n} = \lim_{n \to \infty}{\left(1 +\frac{i\; \alpha}{n}\right)^n} \] As is well known, the definition of the exponential with a real exponent is:
\[e^x = \lim_{n \to \infty}{\left(1 +\frac{x}{n}\right)^n} \] and if we set \( x= i\;\alpha\) we get the definition of the exponential of an imaginary argument:
\[ e^{i \alpha} = \lim_{n \to \infty}{\left(1 +\frac{i\; \alpha}{n}\right)^n} \] So we can finally write the  Euler-Cotes formula⁽²⁾:

\[ e^{i \alpha} = cos \;\alpha + i\;sin\; \alpha \]

As we were able to check in the previous sections, all the typical properties of the exponential function are satisfied.
This formula is an important link between algebra and geometry. Its implications are far-reaching in every field of science.
Its properties are particularly useful in the study of oscillations and waves.

The board on the left shows how the complex number \(\left(1 +\frac{i\; \alpha}{n}\right)^n \) behaves as the parameter n changes.
For each value of n you can also show the points representing the various powers of the number itself:
\[ \dot{p}_n^k=\left(1 +\frac{i\; \alpha}{n}\right)^k \quad k=1, 2, ..., n \] For example, with n = 3 the Gauss plane shows: \( \dot{p}_3^1=\left(1 +\frac{i\; \alpha}{3}\right)^1 \quad \dot{p}_3^2=\left(1 +\frac{i\; \alpha}{3}\right)^2 \quad \dot{p}_3^3=\left(1 +\frac{i\; \alpha}{3}\right)^3 \) It is interesting to see how, as n increases, all the points tend to settle on the unit circle and \(\dot{p}^n_n\) tends to \( \dot{u}_\alpha\). This also gives a visual idea of how, as n increases, \(\left(1 +\frac{i\; \alpha}{n}\right)^n \) approximates better and better the operator of rotation by \(\alpha \) radians (see also section 2.2 below).
The simulation starts with \( \alpha = \pi\), so that, as n increases, it approximates Euler's identity \(e^{i \pi} = -1\) (see section 2.3). Of course you can change \( \alpha\) by moving the point on the circle.
[ Tip: click the point n on the slider and then move it with the ← → keys ]

Considering the complex conjugate of \( \dot{u}_\alpha\), \( \dot{u}^*_\alpha = cos \; \alpha - i\; sin\; \alpha\), we arrive at:
\[ e^{-i \alpha} = \lim_{n \to \infty}{\left(1 -\frac{i\; \alpha}{n}\right)^n} = cos \; \alpha - i\;sin\;\alpha \]
and so:

\[ e^{-i \alpha} = cos \;\alpha - i\;sin \alpha \]

From the formulas above it follows immediately that:
\[ \frac{e^{i \alpha} + e^{-i \alpha}}{2} = cos \;\alpha \quad \quad \frac{e^{i \alpha} - e^{-i \alpha}}{2 i} = sin \alpha \]
Let us conclude by summing up two fundamental properties of the exponential with an imaginary exponent:

a) product of exponentials: \[ e^{i \alpha}\,e^{i \beta} = e^{i \alpha + i \beta}= e^{i (\alpha+\beta)} \] b) n-th order derivative of the exponential: \[ \frac{d^{(n)} e^{i \alpha}}{d \alpha^n} = i^n\; e^{i \alpha}\]

2.2 - The rotation operator \( e^{i \alpha} \)

If \(\alpha = \frac{\pi}{2}\) we have \( e^{i \frac{\pi}{2}} = cos \;\frac{\pi}{2} + i\;sin \frac{\pi}{2}= 0 + i \;1 = i\) and so:

\[ e^{i \frac{\pi}{2}} = i\]

This last formula confirms that \( e^{i \frac{\pi}{2}}\) can be interpreted as an operator of rotation by \(\frac{\pi}{2}\) radians.
Multiplying any complex number by \( e^{i \frac{\pi}{2}}\) is equivalent to multiplying it by i , and therefore to increasing its argument by \(\frac{\pi}{2}\) radians. This means rotating the vector that represents it by 90° counterclockwise.
In the same way, and more generally, \( e^{i \alpha} \) produces a rotation of \(\alpha \) radians. If, for example, \(\dot{z} = z\; \angle \; \beta\), we have: \[ e^{i \alpha}\,\dot{z}= z\; \angle \; \beta + \alpha\]

2.3 - Euler's identity

If \(\alpha = \pi\), then \( e^{i \pi} = cos \;\pi + i\;sin \pi = -1\), and Euler's formula takes the form of the famous Euler's identity,

\[ e^{i \pi} + 1 = 0\]

It is a kind of totem of mathematical knowledge, because it contains the five fundamental numbers:

0	zero, which makes modern positional notation possible;
1	the first number in the sequence of  natural numbers;
π	the ratio of a circle's circumference to its diameter;
e	Euler's number, the base of natural logarithms;
i	the imaginary unit;
+,x,=	the basic operations of multiplication and addition, and the equals sign;

moreover, it relates geometry and algebra through \(\pi\)⁽³⁾, a fundamental number in Euclidean geometry, and \( i\), the imaginary unit.

2.4 - Exponential form of complex numbers

The considerations above lead us to a new form for complex numbers: the exponential form.
Since every complex number can be written in the form \( \dot{z}= r_z\; (cos \; \alpha + i\; sin\; \alpha) \), Euler's formula allows us to write every complex number in the so-called exponential form:

\[\dot{z} = r_z\; e^{i \alpha}\]

where r_z is the modulus of the complex number and α its argument.
Its complex conjugate is:

\[\dot{z}^* = r_z\; e^{-i \alpha}\]

Note that \[\dot{z}\,\dot{z}^* = r_z^2\; e^{i (\alpha - \alpha)} = r_z^2\; e^{i 0} = r_z^2\] where by definition \( e^{i 0} = 1\).

2.5 - Operations with complex numbers in exponential form 

Given two complex numbers in exponential form, \(\dot{z} = r_z\; e^{i \alpha}\) and \(\dot{w} = r_w\; e^{i \beta}\) the multiplication operation becomes:

\[\dot{z}\,\dot{w} = r_z\; e^{i \alpha}\,r_w\; e^{i \beta} = r_z r_w\; e^{i (\alpha + \beta)} \]

The reciprocal of a complex number, for example \(\dot{w}\), is given by:

\[ \dot{w}^{-1} = \frac{1}{ r_w\; e^{i \beta}} = \frac{1}{r_w}\; e^ {-i\beta} \]

and consequently division between complex numbers in exponential form is:

\[\frac{\dot{z}}{\dot{w}} = \frac{r_z}{ r_w} e^{i (\alpha - \beta)} \]

Raising to the n-th power, with integer n, follows directly from multiplication:

\[\dot{z}^n = r_z^n\; e^{i n \alpha} \]

De Moivre's formula can easily be derived from this last one.

As for taking the n-th root:

\[\sqrt[n]{\dot{z}} = \sqrt[n]{r_z}\; e^{i \frac{\alpha + 2 k \pi}{n}} \]

with the integer k = 0, 1, 2, ...

It should now also be clear that, all things considered, the polar form can be thought of as a shorthand for the exponential notation, in which only the modulus and the argument are shown: \[\dot{z} = r_z\; e^{i \alpha} \quad \Leftrightarrow \quad r_z\; \angle \alpha \] so the way operations are carried out in polar form finds further justification in the corresponding operations in exponential form.

3 - Complex powers and logarithms

We now face the problem of defining powers with a complex base and exponent, and the complex logarithm. Given the three complex numbers \(\dot{w} = x + i\;y\), \(\dot{s} = a + i\;b\) and \(\dot{z} = c + i\;d\), we would like to give a meaning to expressions such as \(\dot{w}^{\dot{s}} = (x + i\;y)^{(a + i\;b)}\) and \(\dot{w} = ln(\dot{z})\).

3.1 - Why we need logarithms

We know well that, in the real field, raising a number \( a \) to the integer power n simply means multiplying \( a \) by itself n times.
If we do not want to restrict ourselves to integer exponents and wish to raise \( a \) to the power \( q \), with rational \( q \), we can remember that \( q \) can be written as the ratio of two coprime integers m and n, with n ≠ 0: \( q = \frac{m}{n} \). We must also require \( a\; >\; 0 \) to avoid problems when n is even. Then \( a^q = a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m} \), and in this case too we have a simple answer.
Things get a little harder if the exponent is an irrational or transcendental real number: for example, how can we get the value of \(a^{ \sqrt{2}} = a^{1.41421...} \)?
This is mainly a computational problem, whose difficulty depends on the level of precision we want. Since we know how to compute a power with a rational exponent, we can approximate \(a^{ \sqrt{2}} \approx a^{\frac{14}{10}} = \sqrt[10]{a^{14}} \). For greater precision we would certainly compute \(a^{ \sqrt{2}} \approx a^{\frac{141}{100}} = \sqrt[100]{a^{141}} \), and so on, but we immediately realize how the effort required by the calculation (without the sophisticated tools available today) can quickly become unbearable. Precisely for this reason, at the dawn of the 17th century, when astronomical calculations (in the true sense of the word) kept the scientists of the time busy for weeks and months, logarithms were "invented".

3.2 - Logarithms

Logarithms, introduced by the Scottish mathematician and physicist John Napier in 1614, served as real calculating tools until relatively recent times (in 1972 Hewlett-Packard launched the first scientific pocket calculator, the HP-35, which marked the beginning of the gradual abandonment of the slide rule, which was based precisely on the properties of logarithms). What was the reason for such a great and lasting success? Essentially, the fact that logarithms turn products into sums, powers into products and roots into divisions.
Conceptually, however, the importance of the logarithm goes far beyond the way it can make calculations easier.
The logarithm to base b of a number x  is the exponent to which b must be raised to get x, so if \(x = b^u\), then \(u = log_b (x)\),
that is, u (the exponent) is the logarithm to base b of x, and so \(x = b^{log_b(x)}\).
As a function, the logarithm is the inverse function of the exponential: \(log_b (b^x) = x\)
The properties that make logarithms such a powerful calculating tool are the following (x and y real):

I	\(log_b (x\,y) = log_b(x)+log_b(y)\)	turns a product into a sum   (1' )
II	\(log_b (\frac{x}{y}) = log_b(x)-log_b(y)\)	turns a quotient into a difference
III	\(log_b (x^y) = y\; log_b(x)\)	turns a power into a product (4)
IV	\(log_b (\sqrt[y]{x}) = \frac{1}{y}\; log_b(x)\)	turns taking a root into a division

As is well known, the base normally used for logarithms is Euler's number \(e = 2.718281828459...\)
The logarithm to base \(e\) is commonly written ln and is called the natural logarithm: \(ln(x) = log_e(x)\), because this base turns out to be the most convenient one for computing logarithms.
Briggs' (common) logarithms use base 10. They are written with the symbol log, leaving the base implied, so \(log(x) = log_{10}(x)\).

3.3 - The complex logarithm

Euler's formula allows us to write any complex number in the form:
\[\dot{z} = r_z \; e^{i \alpha+2 k i \pi} \quad \quad k \in \mathbb{Z} \] If, by extension, we apply the definitions of logarithm and exponential used in the real field, and we have \(\dot{z} = e^{\dot{w}} \), then \(\dot{w} \) is the complex natural logarithm of \(\dot{z} \):
\[\dot{w} = ln(\dot{z}) = ln(r_z \; e^{i \alpha}) = ln(r_z) + ln(e^{i \alpha}) \] where we have applied property I of logarithms. But \(ln(e^{i \alpha}) = i \alpha \), and so:
\[\dot{w} = ln(\dot{z}) = ln(r_z \; e^{i \alpha}) = ln(r_z) + i \alpha \] We notice that the imaginary part of the logarithm is precisely the argument α of the number itself. But, as pointed out above for the exponential representation, any increase (or decrease) of the argument by multiples of 2π radians represents the same number.
This ambiguity (to remind us of it, we use the notation \( Ln \) instead of \( ln \)) also affects the logarithm, so that: \[\dot{w} = Ln(\dot{z}) = ln(r_z) + i Arg(\dot{z}) = ln(r_z) + i (\alpha + 2 k \pi) \quad k \in \mathbb{Z}\] where \(ln(r_z)\) is clearly the natural logarithm of the real number \(r_z\).
If in the definition of the logarithm we use the principal value of the argument,  α = arg(z)  ( −π <  α < π ),
we write:
\[\dot{w} = ln(\dot{z}) = ln(r_z) + i \; arg(\dot{z}) = ln(r_z) + i \alpha \quad with \quad -\pi < \alpha < \pi \]
We still have to say something about the exponential of a complex number; suppose \(\dot{w} = x + iy \) \[e^\dot{w} = e^{x+iy} = e^x e^{iy} = e^x (cos\; y + i\; sin\; y) \]
It can be shown ⁽⁵⁾ that this is equivalent to:

\[e^\dot{w} = \lim_{n \to \infty} \left( 1 + \frac{x+ iy}{n}\right)^n = \lim_{n \to \infty} \left( 1 + \frac{\dot{w}}{n}\right)^n \]

3.4 - Complex powers

We now face the problem of defining powers with a complex base and exponent.
Given the two complex numbers \(\dot{w} = x + iy \) and \(\dot{s} = c + id \), we would like to give a meaning to the operation \(\dot{w}^{\dot{s}} = (x + iy)^{c+id} \) Using, by extension, property III of logarithms, we can write:
\[\dot{w}^{\dot{s}} = e^{ln(\dot{w}^{\dot{s}})}= e^{\dot{s}ln(\dot{w})} \] However, operations involving complex logarithms require great care, precisely because of their multiple values.
For example, while the relation \(e^{ln(\dot{w})} = \dot{w} = r \angle \beta \) is always true, since:
\[e^{ln(\dot{w})} = e^{ln(r)+i\beta} = e^{ln(r)} e^{i\beta} = r e^{i\beta} = \dot{w} \] the same does not generally hold for \(ln(e^{\dot{w}}) = \dot{w} \); indeed:

\(e^{\dot{w}} = e^x e^{iy} \)

\(ln(e^{\dot{w}}) = ln(e^x e^{iy}) = ln(e^x) + ln(e^{iy})= x + i(y + 2k\pi) \quad k \in \mathbb{Z} \)




Books by the author and further reading

Books by the author


Rev. October-November 2026