How is Euler's formula derived using Taylor series?

Euler's formula is derived by substituting z = ix into the Taylor series expansion of e^z, simplifying the powers of the imaginary unit i, and separating the resulting series into real and imaginary parts corresponding to the Maclaurin series of cos(x) and sin(x).

Euler's Formula Proof – Derivation Using Taylor Series

The proof of Euler's formula using Taylor series is presented below.

In what follows, \( C \) represents the constant of integration where applicable.

Derivation of Euler's Formula

Referring to the tables of mathematical formulas, the Taylor series expansion of \( e^z \) is given by:

\[ e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \frac{z^4}{4!} + \frac{z^5}{5!} + \frac{z^6}{6!} + \frac{z^7}{7!} + \dots + \frac{z^n}{n!} + \dots \]

Let \( z = ix \), where \( x \) is a real number and \( i = \sqrt{-1} \) is the imaginary unit:

\[ e^{ix} = 1 + (ix) + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \dots \]

Expanding the powers of \( ix \):

\[ e^{ix} = 1 + ix + i^2 \frac{x^2}{2!} + i^3 \frac{x^3}{3!} + i^4 \frac{x^4}{4!} + i^5 \frac{x^5}{5!} + i^6 \frac{x^6}{6!} + i^7 \frac{x^7}{7!} + \dots + i^n \frac{x^n}{n!} + \dots \]

Using the cyclic properties of powers of \( i \):

\( i^2 = -1 \)
\( i^3 = i^2 \cdot i = -i \)
\( i^4 = i^3 \cdot i = 1 \)
\( i^5 = i^4 \cdot i = i \)
\( i^6 = i^5 \cdot i = -1 \)
\( i^7 = i^6 \cdot i = -i \)

Substituting these values back into the expansion:

\[ e^{ix} = 1 + ix - \frac{x^2}{2!} - i\frac{x^3}{3!} + \frac{x^4}{4!} + i\frac{x^5}{5!} - \frac{x^6}{6!} - i\frac{x^7}{7!} + \dots \]

Grouping the real terms together and factoring out \( i \) from the imaginary terms gives equation \( (I) \):

\[ e^{ix} = \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots\right) + i\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots\right) \quad (I) \]

From the tables of mathematical formulas, the Taylor series for \( \sin x \) and \( \cos x \) are:

\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots \] \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots \]

Substituting these series into equation \( (I) \), the real part corresponds to \( \cos x \) and the imaginary part corresponds to \( \sin x \):

\[ e^{ix} = \cos x + i \sin x \]

Derivation of Euler's Formula

More References and Links