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Could you provide a proof of Euler's formula: $e^{it}=\cos t +i\sin t$ ?


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Based on one of the tags, I think you mean $\exp(i t)=\cos(t)+i\sin(t)$, but you need to make it explicit; many mathematical objects carry Euler's name. –  J. M. Aug 28 '10 at 4:19
@J. Mangaldan: Yes, maybe that's the one but there are plenty Euler formulas to choose from :) –  AD. Aug 28 '10 at 4:34
Actually, it is common to define $e^{it}$ using your equation. If something is to be proved we must start by asking what we know about the involved parameters, so how is your definition of $e^{it}$? Do you use a series or some other limit process? –  AD. Aug 28 '10 at 20:54

10 Answers 10

up vote 41 down vote accepted

Assuming you mean $e^{ix}=\cos x+i\sin x$, one way is to use the MacLaurin series for sine and cosine, which are known to converge for all real x in a first-year calculus context, and the MacLaurin series for $e^z$, trusting that it converges for pure-imaginary z since this result requires complex analysis.

The MacLaurin series: $$\begin{align} \sin x&=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots \\ \cos x&=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots \\ e^z&=\sum_{n=0}^{\infty}\frac{z^n}{n!}=1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots \end{align}$$

Substitute $z=ix$ in the last series: $$\begin{align} e^{ix}&=\sum_{n=0}^{\infty}\frac{(ix)^n}{n!}=1+ix+\frac{(ix)^2}{2!}+\frac{(ix)^3}{3!}+\cdots \\ &=1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}-\cdots \\ &=1-\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots +i\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots\right) \\ &=\cos x+i\sin x \end{align}$$

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This is by far the easiest proof to follow. –  Noldorin Aug 29 '10 at 12:16
@Noldorin: yes, but it gives no intuition. The real mystery here is why the RHS should satisfy the identity a(x+y) = a(x) a(y) and this proof gives no insight into this. Of course this is fundamentally a geometric statement about rotation, and a good proof of Euler's formula should have a clear connection to these geometric ideas. –  Qiaochu Yuan Aug 29 '10 at 13:22
+1 This is the way Euler originally proved it (though @Noldorin I think my proof is easier to follow :) –  BlueRaja - Danny Pflughoeft Aug 29 '10 at 17:19
@Quiaochu: Yes, true. Unfortunately, I (among others) aren't familiar enough with group theory (Lie groups specifically) to appreciate it fully. From a fundamental perspective, your proof would seem like the most insightful; from a historical/simple perspective, possibly this one is. –  Noldorin Aug 29 '10 at 20:33

Proof: Consider the function $f(t) = e^{-it}(\cos t + i \sin t)$ for $t \in \mathbb{R}$. By the quotient rule \begin{eqnarray} f^{\prime}(t) = e^{-i t}(i \cos t - \sin t) - i e^{-i t}(\cos t + i \sin t) = 0 \end{eqnarray} identically for all $t \in \mathbb{R}$. Hence, $f$ is constant everywhere. Since $f(0) = 1$, it follows that $f(t) = 1$ identically. Therefore, $e^{it} = \cos t + i \sin t$ for all $t \in \mathbb{R}$, as claimed.

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This is definitely the most elegant proof! –  Abramo Mar 10 '12 at 23:50

Let $\mathbf{A}$ be an $n \times n$ matrix. Recall that the system of differential equations

$$\mathbf{x}' = \mathbf{Ax}$$

has the unique solution $\mathbf{x} = e^{\mathbf{A}t} \mathbf{x}(0)$, where $\mathbf{x}$ is a vector-valued differentiable function and $e^{\mathbf{A}t}$ denotes the matrix exponential. In particular, let $\mathbf{J} = \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right]$. Then the system of differential equations

$$x' = y, y' = -x$$

with initial conditions $x(0) = 1, y(0) = 0$ has the unique solution $\left[ \begin{array}{cc} x \\ y \end{array} \right] = e^{\mathbf{J}t} \left[ \begin{array}{cc} 1 \\ 0 \end{array} \right]$. On the other hand, the above equations tell us that $x'' = -x$ and $y'' = -y$, and we know that the solutions to this differential equation are of the form $a \cos t + b \sin t$ for constants $a, b$. By matching initial conditions we in fact find that $x = \cos t, y = \sin t$. Now verify that on vectors multiplying by $\mathbf{J}$ has the same effect as multiplying a complex number by $i$, and you obtain Euler's formula.

This proof has the following attractive physical interpretation: a particle whose $x$- and $y$-coordinates satisfy $x' = y, y' = -x$ has the property that its velocity is always perpendicular to and has proportional magnitude to its displacement. But from physics lessons you know that this uniquely describes particles which move in a circle.

Another way to interpret this proof is as a description of the exponential map from the Lie algebra $\mathbb{R}$ to the Lie group $\text{SO}(2)$. Euler's formula generalizes to quaternions, and this in turn can be thought of as describing the exponential map from the Lie algebra $\mathbb{R}^3$ (with the cross product) to $\text{SU}(2)$ (which can then be sent to $\text{SO}(3)$). This is one reason it is convenient to use quaternions to describe 3-d rotations in computer graphics; the exponential map makes it easy to interpolate between two rotations.

Edit: whuber's answer reminded me of the following excellent graphic.

This is what is happening geometrically in whuber's answer, and is essentially what happens if you apply Euler's method to the system of ODEs I described above. Each step is a plot of the powers of $1 + \frac{i}{N}$ up to the $N^{th}$ power.

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Is there the possibility of circular logic here? You are using too many cannons... one of which might well require the result you are trying to prove. –  Aryabhata Aug 28 '10 at 18:43
@Moron: everything I said requires only real-variable techniques except the last step, which is trivial. In particular one can show that the solutions to x'' = -x are linear combinations of sine and cosine directly; this doesn't require the theory of the characteristic polynomial, if that's what you're thinking of. I would appreciate if you could be more specific. –  Qiaochu Yuan Aug 28 '10 at 19:16
@Qiaochu: I was just wondering if there could be a case of circular logic. I had nothing specific in mind. Euler's formula is quite a fundamental result, and we never know where it could have been used. I don't expect one to know the proof of every dependent theorem of a given result. But anyway, you seem to have justification, so I won't bother you :-) –  Aryabhata Aug 29 '10 at 7:22
Thank you for the graphic! –  whuber Aug 29 '10 at 14:37
@Moron: the principal part of Qiaochu's answer--namely, that Euler's formula can be understood as the exponential map from R to SO(2)--is a modern way of expressing the elementary construction described in my answer. An advantage of the modern approach is its far reach, as hinted at by the application to SU(20; a possible disadvantage is that to the uninitiated it appears to obscure the basic geometric idea. But there's no "circular logic." –  whuber Aug 29 '10 at 15:07

One could provide answers based on a wide range of definitions of $\exp$, $\cos$, and $\sin$ (e.g., via differential equations, power series, Lie theory, inverting integrals, infinite sums, infinite products, complex line integrals, continued fractions, circular functions, and even Euclidean geometry) as well as offering Euler's formula up as a tautology based on a definition. But let's consider where in one's education this question arises: it's usually well before most of these other concepts are encountered. The complex numbers have just been introduced; the Argand diagram is available; $\exp(x)$ is most likely defined as the limiting value of $(1 + x/n)^n$, and $\cos$ and $\sin$ are defined as circular functions (that is, as arclength parameters for coordinates of the unit circle). So the question deserves a response in this context using mathematics accessible to someone at this level.

Accordingly, I propose that we interpret $\exp(i x)$ as the limit of $(1 + i x/n)^n$, because at least one understands how to compute the latter (as iterated multiplication of 1 by $1 + i x/n$), one at least vaguely intuits what the limit of a sequence of points in the complex plane might mean, and one has learned that $\cos(\theta) + i \sin(\theta)$ is quite literally the complex number plotted at $\left( \cos(\theta), \sin(\theta) \right)$ in the plane.

It seems natural to evaluate the limit by looking at its modulus and argument separately. The modulus is relatively easy: because $|1 + i x/n| = \sqrt { 1 + (x/n)^2 }$, the modulus of its $n$th power equals $\left( 1 + (x/n)^2 \right) ^{n/2}$. We can exploit the one limit assumed to be known:

$\left( 1 + (x/n)^2 \right) ^{n/2} = \left( \left( 1 + x^2 / n^2 \right) ^ {n^2} \right) ^ {1/{2n}} \simeq \left( \exp ( x^2 ) \right) ^ {1/{2n}} \to 1$.

(There's some hand-waving in the last two steps. This is characteristic of limiting arguments at this level. Those of you who can see where the rigor is lacking also know exactly how to supply the missing steps.) Whence, whatever $\exp( i x )$ might be, we deduce it should lie on the unit circle.

Now we turn our attention to the argument of the limit. The sequence of arguments

$\left( 1 + (x/n)^2 \right) ^{1/2}, \left( 1 + (x/n)^2 \right) ^{2/2}, \ldots, \left( 1 + (x/n)^2 \right) ^{k/2}, \ldots$

obviously is non-decreasing, because it's a geometric sequence with multiplier of 1 or greater. That is, each successive multiplication by $1 + i x / n$ is expanding an original right triangle with vertices at (0,0), (1,0), and (1, $x/n$), but in the limit the amount of expansion is reduced to unity, as we have seen. In the Argand diagram we're just laying out similar copies of this original right triangle, placing one leg of each new copy along the hypotenuse of the previous one. In the limit, the length of the small leg (originally $x/n$) therefore remains constant. These observations imply that near the limit, we can take all these $n$ little triangles to be essentially congruent, whence the length of the path traced out by the succession of images of the small leg must be extremely close to $n (x/n) = x$. This pins down where on the circle $\exp( i x )$ must lie: it is the point reached by traveling (signed) distance $x$ counterclockwise along the circle beginning at (1,0). To anyone exposed to the definition of $\sin$ and $\cos$ in terms of circular functions, it is immediate that the coordinates of this limiting location are $\left( \cos(x), \sin(x) \right)$ and we are done.

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This to my mind is the most intuitive and explanatory argument. (It was a long time after I knew Euler's identity that I discovered this argument, which was the moment I felt I had truly understood Euler's identity on a visceral level.) The same argument is hinted at in Conway and Guy's The Book of Numbers, with a supporting (static) graphic. –  user43208 Sep 1 '13 at 23:38

HINT $\:$ Both $\:\rm e^{ix}\:$ and $\:\rm cos(x) + i \; sin(x) \:$ are solutions of $\;\rm y' = i \; y,\;\; y(0) = 1, \;$ so they are equal by the uniqueness theorem. Alternatively, bisect into even and odd parts the power series for $\:\rm e^{ix} \:,\;$ i.e.

$$\begin{align} \rm f(x) \ \ &=\ \ \rm\frac{f(x)+f(-x)}{2} \;+\; \frac{f(x)-f(-x)}{2} \\ \\ \Rightarrow\quad\quad \rm e^{ix} \ \ &=\ \ \rm\cos(x) \ +\ i \:\sin(x) \end{align}$$

REMARKS 1.$\;$ Uniqueness theorems provide powerful tools for proving equalities for functions that satisfy certain nice differential or difference (recurrence) equations. This includes a large majority of functions encountered in theory and practice. Such ideas lie behind algorithms implemented in computer algebra systems, e.g. search the computational literature using the terms "D-finite" and/or "holonomic" functions. For a less trivial but still easy example of this technique see my recent post which proves the identity

$$\rm \frac{sinh^{-1}(x)}{\sqrt{x^2+1}} \ = \ \ \sum_{k=0}^\infty\ (-1)^k \frac{(2k)!!}{(2k+1)!!} \: x^{2k+1}$$

For a simple discrete example see my post here where I remark that $\rm\; 13 \:|\: 3^{n+1} + 4^{2n-1} =: f_n \;$ follows from $\;\rm f_2 = f_1 = 0 \;$ and the (obvious) fact that $\;\rm f_n \;$ satisfies a monic 2nd order linear recurrence. Namely let $\;\rm S\; f_n := f_{n+1} \;$ be the shift operator. Then $\;\rm S - 3 \;$ kills $\rm\; 3^{n+1} \;$ and $\;\rm S - 16 \;$ kills $\;\rm 4^{2n-1} \;$ therefore $\;\rm (S-3)\:(S-16) = S^2 - 19\: S + 48 \;$ kills their sum $\;\rm f_n \:,\;$ i.e. $\;\rm f_{n+2} = 19\; f_{n+1} - 48\; f_n\;$. Therefore $\:\rm mod\:13:\ f_2 = f_1 = 0 \;\Rightarrow\; f_3 = 19\: f_2 - 48\: f_1 = 0 \;\Rightarrow f_4 = 0 \Rightarrow f_5 = 0 \Rightarrow \cdots\Rightarrow\; f_n = 0\: $. Hence $\:0\:$ is the unique solution of the above recurrence that satisfies the intial conditions $\;\rm f_2 = f_1 = 0. \;$ This is simply an obvious special case of the uniqueness theorem for difference equations (recurrences). Once again, by invoking a uniqueness theorem, we have greatly simplified the deduction of an equality.

Notice that, above, we don't need to know the precise recurrence relation. Rather, we need only know a bound on its degree, so that we know how many initial terms are needed to determine the solution uniquely. In practice, as above, one can often easily derive simple upper bounds on the degree of the recurrence or differential equation - which makes the method even more practical.

2. $\;$ Generalizing the above bisection into even and odd parts, one can employ n'th roots of unity to take arbitrary n-part multisections of power series and generating functions. They often prove handy, e.g.

EXERCISE $\;$ Give elegant proofs of the following

$\quad\quad\rm\displaystyle sin(x)/e^{x} \quad\:$ has every $\rm 4k\;$'th term zero

$\quad\quad\rm\displaystyle cos(x)/e^{x} \quad$ has every $\rm 4k+2\;$'th term zero

See the posts in this thread for various solutions and more on multisections.

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Well this question actually boils down to "How is the complex exponential defined?"

Here is my view of this problem:


$$f(x+iy)= e^{x}(\cos(y)+i\sin(y) ) \,.$$

Then $f$ has continuous partial derivatives $f_x$ and $f_y$, and verifies the Cauchy-Riemann equations, thus it is analytic.

Moreover, for any $z_1,z_2 \in {\mathbb C}$ we have

$$f(z_1+z_2)=f(z_1) f(z_2) \,.$$

Last but not least $f(x)=e^x$ for all $x \in {\mathbb R}$.

In particular we showed that $e^x$ can be extended to an analytic complex function, and the theory tells us that such extension is unique.

Now, since $f(z)$ is the unique analytic extension of $e^x$ to the complex plane, and it also satisfies the exponential relation $f(z_1+z_2)=f(z_1) f(z_2)$, we call this function $e^z$.

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"Combining" the answers from Qiaochu Yuan and Isaac one can also directly evaluate $\begin{bmatrix}x \\ y \end{bmatrix} = e^{\mathbf{J}t} \begin{bmatrix}1 \\ 0 \end{bmatrix}$ with $\mathbf{J}=\begin{bmatrix}0&1 \\ -1&0 \end{bmatrix}$ (note that $\mathbf{J}^2 = - \begin{bmatrix}1&0 \\ 0&1 \end{bmatrix}$) as follows:

$$ e^{\mathbf{J}t} \begin{bmatrix}1 \\ 0 \end{bmatrix} = \left( \mathbf{J}^0 + \frac{t}{1!} \mathbf{J}^1 + \frac{t^2}{2!} \mathbf{J}^2 + \frac{t^3}{3!} \mathbf{J}^3 + \frac{t^4}{4!} \mathbf{J}^4 + \cdots \right) \begin{bmatrix}1 \\ 0 \end{bmatrix} \\ = \left( \begin{bmatrix}1&0 \\ 0&1 \end{bmatrix} + t \begin{bmatrix}0&1 \\ -1&0 \end{bmatrix} - \frac{t^2}{2!}\begin{bmatrix}1&0 \\ 0&1 \end{bmatrix}-\frac{t^3}{3!}\begin{bmatrix}0&1 \\ -1&0 \end{bmatrix}+ \frac{t^4}{4!}\begin{bmatrix}1&0 \\ 0&1 \end{bmatrix}+\cdots\right)\begin{bmatrix}1 \\ 0 \end{bmatrix} \\ = \begin{bmatrix} 1 + 0 - \frac{t^2}{2!} - 0 + \frac{t^4}{4!} + \cdots \\ 0 - t - 0 + \frac{t^3}{3!} + 0 + \cdots \end{bmatrix} \\ = \begin{bmatrix} \cos(t) \\ - \sin(t) \end{bmatrix} $$ The result is a parametrization of a helix curve. That means Eulers formula simply shows how one can parametrize a helix using the exponential function.

The result can also be written like $$ e^{\mathbf{J}t} \begin{bmatrix}1 \\ 0 \end{bmatrix} = \left( \cos(t) + \mathbf{J} \sin(t) \right)\begin{bmatrix} 1 \\ 0 \end{bmatrix} $$ making the connection to the traditional form of Eulers formula manifest.

It should be noted that with $\mathbf{K}=\begin{bmatrix}0&1 \\ 1&0 \end{bmatrix}$ the following relation can be proved in the same way: $$ e^{\mathbf{K}t} \begin{bmatrix}1 \\ 0 \end{bmatrix} = \left( \cosh(t) + \mathbf{K} \sinh(t) \right)\begin{bmatrix} 1 \\ 0 \end{bmatrix} $$

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An easy way out to answer the question is when we try to extend the definition of exponential to complex plane in a "nice" way (read "nice" as holomorphic) we then end up with this definition. And as you are probably aware there is only at most one such extension. So if we want to define our exponential in the complex plane such that the exponential function is holomorphic, and it matches with the conventional exponential function on the real line, we end up with $e^{it} = \cos(t) + i \sin(t)$.

All the other answers I think are circular in some sense.

For instance, how do we know that we can expand $e^{i x}$ as a power series. All we know is that we can expand $e^{x}$ as a power series when $x$ is real. We do not know apriori that the expansion can be carried forward if $x$ happens to be complex.

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Any answer will have to give/assume some definition of what $e^{it}$ means, but that is not the same as assuming the result. For example, taking the unique holomorphic extension of $\exp$ (as you suggest) leads to the power series expansion. But then the formula in question holds based on the series for $\exp$ and the series for the real functions $\sin$ and $\cos$, as shown in Isaac's answer, but this formula is not actually part of the definition. Answers that don't give full justification may be incomplete as far as rigorous proof goes, but that is different from being circular. –  Jonas Meyer Nov 25 '10 at 14:30

How about the Laplace transform of

$$\exp(i t)=\cos(t)+i\sin(t)$$

Let's evaluate the Laplace transform of the two sides of the formula:


Now, let's multiply both sides by $s-i$ :



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I'll copy my answer to this question:

Let $f(x) = \cos(x) + i\cdot \sin(x)$

Then $$\begin{align*}\frac{df}{dx} &= -\sin(x) + i \cdot \cos(x)\\ &= i \cdot f(x) \end{align*}$$

So that $$ ∫(\frac{1}{f(x)}) df = ∫i \cdot dx\\ \ln(f(x)) = ix + C\\ f(x) = e^{ix + C} = \cos(x) + i \cdot \sin(x)\\ $$

Since $f(0) = 1, C = 0$, so: $$ e^{ix} = \cos(x) + i \cdot \sin(x) $$

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There appear to be some gaps here; the most significant one concerns the use of a complex logarithm. You seem to have overlooked the fact there are many more solutions to exp(C) = 1 than merely C = 0. Their existence is an implication of Euler's formula itself. To avoid circular logic, the crucial thing is to make clear what definitions you are using for exp, sin, and cos (and also, in this case, of the complex logarithm). –  whuber Aug 30 '10 at 14:21

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