M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

Euler's Formula for Involutions


Euler's formula for scalar complex numbers is

\begin{equation*} \exp{(iz)} = \cos{(z)} + i \sin{(z)} \end{equation*}

Recall that

\begin{align*} \cos{(z_{k})} &= 1 & \sin{(z_{k})} &= 0 & z_{k} &= 2k \pi & k &\in \mathbb{Z} \\ \cos{(z_{l})} &= -1 & \sin{(z_{l})} &= 0 & z_{l} &= (2l-1) \pi & l &\in \mathbb{Z} \\ \cos{(z_{m})} &= 0 & \sin{(z_{m})} &= 1 & z_{m} &= \frac{(4m+1) \pi}{2} & m &\in \mathbb{Z} \\ \cos{(z_{n})} &= 0 & \sin{(z_{n})} &= -1 & z_{n} &= \frac{(4n-1) \pi}{2} & n &\in \mathbb{Z} \end{align*}

Thus, you have the exponential representations

\begin{align*} 1 &= \exp{[2 k \pi i]} & {-1} &= \exp{[(2l - 1) \pi i]} & i &= \exp{\left[ \frac{(4m+1) \pi i}{2} \right]} & {-i} &= \exp{\left[ \frac{(4n-1) \pi i}{2} \right]} \end{align*}

Now consider taking the matrix exponential of a matrix proportional to the identity matrix \(I\):

\begin{equation*} \exp{(izI)} = I \left[\cos{(z)} + i \sin{(z)}\right] \end{equation*}

Thus, you have the following exponential representations:

\begin{align*} I &= \exp{[2 k \pi i I]} & {-I} &= \exp{[(2l - 1) \pi i I]} & iI &= \exp{\left[ \frac{(4m+1) \pi i}{2}I \right]} & {-i}I &= \exp{\left[ \frac{(4n-1) \pi i}{2}I \right]} \end{align*}

Let the matrix \(A\) be an involution:

\begin{equation*} A^{2} = I \end{equation*}

Then

\begin{equation*} \exp{(iz A)} = I \cos{(z)} + i A \sin{(z)} \end{equation*}

Now you have the following exponential representations:

\begin{align*} I &= \exp{\left[ 2 k \pi i A \right]} & k &\in \mathbb{Z} \\ {-I} &= \exp{\left[ (2 l - 1) \pi i A \right]} & l &\in \mathbb{Z} \\ iA &= \exp{\left[ \frac{1}{2} (4m+1) \pi i A \right]} & m &\in \mathbb{Z} \\ {-iA} &= \exp{\left[ \frac{1}{2} (4n-1) \pi i A \right]} & n &\in \mathbb{Z} \end{align*}

Using the above result, you can have another exponential representation of the identity:

\begin{align*} I &= \exp{\left[ (2k - 1) \pi i I + (2l - 1) \pi i A \right]} & k, l &\in \mathbb{Z} \end{align*}

and two exponential representations of the involution:

\begin{align*} A &= \exp{\left[ \frac{1}{2}(4k-1) \pi i I + \frac{1}{2} (4l+1) \pi i A \right]} & k, l &\in \mathbb{Z} \\ A &= \exp{\left[ \frac{1}{2}(4m+1) \pi i I + \frac{1}{2} (4n-1) \pi i A \right]} & m, n &\in \mathbb{Z} \end{align*}

You can define the principal values as the cases when \(k = l = m = n = 0\).