Skip to the content

Math 2.0

Special functions Proof

Proof of de Moivre's formula from Euler's formula

Builds on Euler's formula Product of exponentials Complex exponential Sine and cosine

Proof

Let $x$ be real. Since $x$ is real, the power series of $\cos x$ and $\sin x$ are the real sine and cosine. So Euler's formula gives $e^{i x} = \cos x + i \sin x$ for every real $x$ .
In the series for the complex exponential at $0$ , every term with $k \ge 1$ is $0$ . So $e^0 = 1$ .
Claim: $(e^{i x})^n = e^{i n x}$ for each integer $n \ge 0$ and each real $x$ . We argue by induction on $n$ .
For $n = 0$ , both sides equal $1$ , by step 2.
Assume the claim for $n$ . By the product of exponentials, $(e^{i x})^{n+1} = e^{i n x} e^{i x} = e^{i (n + 1) x}$ . This proves the claim.
By the product of exponentials and step 2, $e^{i x} e^{-i x} = e^0 = 1$ . So $e^{i x} \neq 0$ and $(e^{i x})^{-1} = e^{-i x}$ . Thus negative powers of $e^{i x}$ are defined.
Let $n = -m$ with $m$ a positive integer. Then $(e^{i x})^n = \bigl((e^{i x})^{-1}\bigr)^m = (e^{-i x})^m$ , by step 6.
Step 3, applied to the real number $-x$ , gives $(e^{-i x})^m = e^{-i m x} = e^{i n x}$ . So $(e^{i x})^n = e^{i n x}$ for every integer $n$ .
By step 1 for $x$ and for $n x$ , $(\cos x + i \sin x)^n = (e^{i x})^n = e^{i n x} = \cos(n x) + i \sin(n x)$ .

Idea

Euler's formula turns $\cos x + i \sin x$ into $e^{i x}$ . Powers of an exponential multiply the exponent, because exponentials turn sums into products. Negative powers work since $e^{-ix}$ is the inverse of $e^{ix}$ .

Reading path

To understand this page from the ground up, read these 111 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof that a complex number times its conjugate is the square of its modulus, Proof of the equality of complex numbers, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof that i squared is minus one, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that the modulus is multiplicative, Proof of the factorial formula for binomial coefficients by induction, Proof of the product of exponentials by the Cauchy product, Proof that the complex numbers form a field, Proof of the binomial theorem by induction, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof of Euler's formula by power series, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product.

Binomial coefficient definition
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Element definition
Factorial definition
Group concept
Imaginary unit object
Integers object
Mathematical induction concept
Ordered field definition
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Proof of Pascal's rule from the recursive definition proof
Complex exponential definition
Power series in the complex plane definition
Real and imaginary parts definition
Proof that i squared is minus one proof
Set definition
Ring definition
Ring of integers statement
Implication definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Supremum definition
Pascal's rule statement
Complex conjugate definition
i squared is minus one statement
Ordered pair definition
Commutative ring definition
Unit of a ring definition
Universal quantifier definition
The rationals form a field statement
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Completeness axiom statement
Proof of the factorial formula for binomial coefficients by induction proof
Cartesian product definition
Proof of the equality of complex numbers proof
Field definition
Subset definition
The rationals form an ordered field statement
Convergent sequence definition
Cauchy product definition
Partial sum definition
Power series definition
Factorial formula for binomial coefficients statement
Binary relation definition
Equality of complex numbers statement
Order on the reals definition
Convergent series definition
Sine and cosine definition
Antisymmetric relation definition
Proof of the binomial theorem by induction proof
Reflexive relation definition
Transitive relation definition
Proof that the complex numbers form a field proof
Sum of a series definition
Absolute convergence definition
Binomial theorem statement
Preorder definition
The complex numbers form a field statement
Partial order definition
Total order definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Triangle inequality statement
Absolute value is multiplicative statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Modulus of a complex number definition
Proof that convergent sequences are bounded proof
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Proof that a complex number times its conjugate is the square of its modulus proof
Limit of a sum of sequences statement
Convergent sequences are bounded statement
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Monotone convergence theorem for sequences statement
A complex number times its conjugate statement
Proof of the limit of a product of sequences proof
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that the modulus is multiplicative proof
Limit of a product of sequences statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Modulus is multiplicative statement
Proof of Euler's formula by power series proof
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Euler's formula statement
Cauchy product of absolutely convergent series statement
Proof of the product of exponentials by the Cauchy product proof
Product of exponentials statement
Proof of de Moivre's formula from Euler's formula proof

Where this leads

30 pages build on this one.

Further reading

Web https://en.wikipedia.org/wiki/De_Moivre%27s_formula, Relation to Euler's formula

About this page

Tag	`21PKVV5S4Y`
URI	`m2://21PKVV5S4Y`
Kind	proof
Writing	W0 stub
MSC	33B10
Proves	de Moivre's formula
Uses	Euler's formula, Product of exponentials, Complex exponential, Sine and cosine
Changed	4 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.