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Being one the most astounding formulas in mathematics, Euler’s identity is popularly called God’s equation. Some people also go as far as calling it the mathematical equivalent of Da Vinci's Mona Lisa or Michaelangelo's David.

Named after Leonhard Euler, the formula establishes the deep relationship between trigonometric functions and complex exponential functions.

According to the formula, for any real number x,

In the above formula, e is the base of the natural logarithm, i the imaginary unit. Cos and Sin are trigonometric functions (the arguments are to be taken in radians and not degrees.). The formula applies even if x is a complex number.

Particularly with x = π, or half a turn around the circle,

e^iπ = cos π + i sin π

Since cos π = -1 and sin π = 0,

It can be deduced that e^iπ = -1 + i0 which brings us to the identity

The identity successfully links five fundamental mathematical constants: 1. The number 0(the additive identity). 2. The number 1(the multiplicative identity). 3. The number pi (3.14159265…). 4. The number e (base of all natural logarithms, which occurs widely in mathematics and scientific analysis). 5. The number i (the imaginary unit of the complex numbers)

The formula describes two equivalent ways to move in a circle.

One of its major applications is that in the complex number theory.

The interpretation of the function eix can be that it traces out the unit circle in the complex number plane while x ranges through the real numbers. x in this case refers to the angle that any line that connects the origin with any point on the circle makes with the positive real axis (being measured in radians counter clockwise).

Points in the complex plane are represented by complex numbers that are written in cartesian coordinates. Euler’s formula provides a means of conversion between polar coordinates and cartesian coordinates. The number of terms is reduced from two to one by polar form, which simplifies the mathematics for multiplication or powers of complex numbers. Therefore any complex number, z = x + iy can be written as

In the above, x = Re {z} the real part y = Im{z} the imaginary part the magnitude of z ø = arg z = atan2(y , x) in this is the argument of z or in other words is the angle (clockwise and in radians) between x-axis and the vector z.

Using the Euler’s formula on the derived formula, the logarithm of a complex number can be defined. For this we need to take the definition of the logarithm (as an inverse operator of exponentiation):

and

both being valid for any complex numbers a and b.

Therefore it can be written that

for any number z ≠ 0. On taking the logarithm on both sides, it can be seen that

This in fact can also be used to define the complex logarithm. The logarithm of a complex number is therefore a multi valued function as φ is multi valued.

Finally, the other exponential law

which can be seen to hold for all integers k, together with Euler's formula, implies severaltrigonometric identities as well as de Moivre's formula.

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