Euler’s Formula for Complex Numbers

Interpretation of Euler’s formula

Euler’s formula tells us the following:

$latex {{e}^{ix}}=\cos(x)+i~\sin(x)$

In this formula we have:

• x is a real number
• e is the base of the natural logarithm (approximately 2,718…)
• i is the imaginary unit (square root of -1)

Euler’s formula establishes the relationship between trigonometric functions and exponential functions. This formula can be thought of geometrically as a way of relating two representations of the same complex number in the complex plane.

The following are some key values ​​in Euler’s formula that correspond to important points on the unit circle:

• For $latex x=0$, we have $latex {{e}^{0}}=\cos(0)+i ~ \sin(0)$, which results in $latex 1=1$. We know that an angle of 0 on the unit circle is equal to 1 on the real axis.

• For $latex x=1$, we have $latex {{e}^{i}}= \cos (1) +i ~ \sin(1)$. This suggests that $latex {{e}^i}$ is the point on the unit circle with an angle of 1 radian.

• For $latex x=\frac{\pi}{2}$, we have $latex {{e}^{i \frac{\pi}{2}}}=\cos (\frac{\pi}{2})+i ~ \sin(\frac{\pi}{2})$. This result is frequently applied in physics.

• For $latex x=\pi$, we have $latex {{e}^{i \pi}}=\cos(\pi)+i ~ \sin(\pi)$, which results in $latex {{e}^{i \pi}}=-1$. The result is the Euler identity.

• For $latex x=2 \pi$, we have $latex {{e}^{i (2\pi)}}=\cos(2\pi)+i ~\sin(2 \pi)$, which results in $latex {{e}^{i (2 \pi)}} = 1$. Similar to the result of using 0.

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Euler’s identity

Euler’s identity is often considered the most beautiful equation in mathematics. Euler’s identity is written as follows:

$latex {{e}^{i\pi}}+1=0$

This equation contains the five most important constants in mathematics:

• The additive identity, 0
• Unity, 1
• The constant $latex \pi$ (quotient of a circle to its radius)
• The base of the natural logarithm e
• The imaginary unit i

Here, we have three different types of numbers: integers, irrational numbers, and imaginary numbers. We also have three of the basic mathematical operations: addition, multiplication, and exponentiation.

Euler’s identity is obtained by starting with Euler’s formula:

$latex {{e}^{ix}}= \cos(x)+i ~ \sin(x)$

and by using $latex x= \pi$ and moving the resulting -1 to the left side.

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Applications of Euler’s formula

Euler’s formula can be used to facilitate the computation of operations with complex numbers, trigonometric identities, and even the integration of functions. With Euler’s formula, we can write complex numbers in their exponential form, write alternate definitions of important functions, and obtain trigonometric identities.

Complex numbers in exponential form

We know that a complex number can be written in Cartesian coordinates like $latex a+ bi$, where a is the real part and b is the imaginary part..

We also know that the same complex number can be expressed in polar coordinates as $latex r(\cos(\theta+i~\sin(\theta))$, where r is the magnitude of the number, and $latex \theta$ is its angle with respect to the positive x-axis.

Thanks to Euler’s formula, all complex numbers can be written as exponentials as follows:

$latex z=r(\cos(\theta+i~\sin(\theta))=r{{e}^{i\theta}}$

The exponential form of complex numbers makes multiplying complex numbers much easier. For example, given two complex numbers $latex z_{1}=r_{1}{{e}^{i\theta _{1}}}$ y $latex z_{2}=r_{2}{{e}^{i\theta _{2}}}$, we can multiply them as follows:

$latex z_{1}z_{2}=r_{1}{{e}^{i\theta _{1}}}\times r_{2}{{e}^{i\theta _{2}}}$

$latex =r_{1}r_{2}{{e}^{i(\theta _{1}+\theta _{2})}}$

Similarly, we can divide two complex numbers by dividing their magnitudes and subtracting their angles.

Alternate definitions of important functions

Euler’s formula can also be used to obtain alternate definitions for different important functions such as trigonometric functions and hyperbolic functions.

For example, it is also possible to use Euler’s formula to derive a similar equation for the opposite angle $latex -x$:

$latex {{e}^{-ix}}=\cos(x)-i~\sin(x)$

This equation, along with the original Euler’s formula, constitutes a system of equations from which we can solve for both the sine function and the cosine function. For example, by subtracting the equation $latex {{e}^{-ix}}$ from the equation $latex {{e}^{ix}}$, the cosines cancel out and after dividing by $latex 2i$, we get an expression for the sine function:

$latex \sin(x)=\frac{{{e}^{ix}}-{{e}^{-ix}}}{2i}$

Similarly, when adding the two equations, the sines cancel and after dividing by 2, we obtain an expression for the cosine function:

$latex \cos(x)=\frac{{{e}^{ix}}+{{e}^{-ix}}}{2}$

The tangent function can be obtained by dividing the sine by the cosine. Using similar methods, expressions for hyperbolic functions and other important functions can also be obtained.

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See also

Interested in learning more about complex numbers? Take a look at these pages:

• What are Complex Numbers?
• Imaginary and Complex Numbers with Exponents
• Operations with Complex Numbers