Euler's formula states for any real number x,

where e is the base of the natural logarithm, i is the imaginary unit that satisfies the equation i2 = -1, and sine and cosine are trigonometric functions.

The Euler formula can be demonstrated using a series expansion.

References

Greenberg, Michael D. (1998), Advanced Engineering Mathematics, Prentice-Hall (New Jersey).

Ogata, Katsuhiko (1998), System Dynamics (Third Edition), Prentice-Hill (New Jersey).

Spiegel, Murray R. (1964), Schaum’s Outline of Theory and Problems of Complex Variables, McGraw-Hill (New York).