Consider the following equation:
\begin{equation*}
\mu^{4} = 1
\end{equation*}
The four solutions are
\begin{align*}
\mu_{0} &= 1, & \mu_{1} &= i, & \mu_{2} &= -1, & \mu_{3} &= -i
\end{align*}
Now consider the following four functions:
\begin{align*}
f_{0}(z) &\equiv \exp{(\mu_{0} z)} = \exp{( z)} \\
f_{1}(z) &\equiv \exp{(\mu_{1} z)} = \exp{( iz)} \\
f_{2}(z) &\equiv \exp{(\mu_{2} z)} = \exp{( -z)} \\
f_{3}(z) &\equiv \exp{(\mu_{3} z)} = \exp{( -iz)}
\end{align*}
First, you write \(f_{0}\) as an infinite series, split into four sums:
\begin{equation*}
f_{0}(z) = \sum_{n = 0}^{\infty} \frac{1}{\Gamma(n + 1)} z^{n} = p_{0}(z) + p_{1}(z) + p_{2}(z) + p_{3}(z)
\end{equation*}
Here, the \(p\)-functions are given by
\begin{align*}
p_{0}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(4n + 1)} z^{4n} \\
p_{1}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(4n + 2)} z^{4n+1} \\
p_{2}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(4n + 3)} z^{4n+2} \\
p_{3}(z) &= \sum_{n = 0}^{\infty} \frac{1}{\Gamma(4n + 4)} z^{4n+3}
\end{align*}
You can also write the other \(f\)-functions in terms of the \(p\)-functions:
\begin{align*}
f_{1}(z) &= p_{0}(z) + i p_{1}(z) - p_{2}(z) - i p_{3}(z) \\
f_{2}(z) &= p_{0}(z) - p_{1}(z) + p_{2}(z) - p_{3}(z) \\
f_{3}(z) &= p_{0}(z) - i p_{1}(z) - p_{2}(z) + i p_{3}(z)
\end{align*}
Note that this definition of the \(p\)-functions can be stated as a matrix equation:
\begin{equation*}
\begin{bmatrix}
f_{0}(z) \\
f_{1}(z) \\
f_{2}(z) \\
f_{3}(z)
\end{bmatrix} = \begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & i & -1 & -i \\
1 & -1 & 1 & -1 \\
1 & -i & -1 & i
\end{bmatrix} \begin{bmatrix}
p_{0}(z) \\
p_{1}(z) \\
p_{2}(z) \\
p_{3}(z)
\end{bmatrix}
\end{equation*}
The matrix that appears here corresponds to the non-unitary 4-dimensional discrete Fourier transform. Inverting this matrix gives
\begin{equation*}
\begin{bmatrix}
p_{0}(z) \\
p_{1}(z) \\
p_{2}(z) \\
p_{3}(z)
\end{bmatrix} = \frac{1}{4} \begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & -i & -1 & i \\
1 & -1 & 1 & -1 \\
1 & i & -1 & -i
\end{bmatrix} \begin{bmatrix}
f_{0}(z) \\
f_{1}(z) \\
f_{2}(z) \\
f_{3}(z)
\end{bmatrix}
\end{equation*}
More explicitly:
\begin{align*}
p_{0}(z) &= \frac{f_{0}(z) + f_{1}(z) + f_{2}(z) + f_{3}(z)}{4} = \frac{\cosh{(z)} + \cos{(z)}}{2} \\
p_{1}(z) &= \frac{f_{0}(z) - if_{1}(z) - f_{2}(z) + if_{3}(z)}{4} = \frac{\sinh{(z)} + \sin{(z)}}{2} \\
p_{2}(z) &= \frac{f_{0}(z) - f_{1}(z) + f_{2}(z) - f_{3}(z)}{4} = \frac{\cosh{(z)} - \cos{(z)}}{2} \\
p_{3}(z) &= \frac{f_{0}(z) + if_{1}(z) - f_{2}(z) - if_{3}(z)}{4} = \frac{\sinh{(z)} - \sin{(z)}}{2}
\end{align*}
The parity properties are as follow:
\begin{align*}
p_{0}(-z) &= p_{0}(z), & p_{1}(-z) &= -p_{1}(z), & p_{2}(-z) &= p_{2}(z), & p_{3}(-z) &= -p_{3}(z) \\
p_{0}(iz) &= p_{0}(z), & p_{1}(iz) &= ip_{1}(z), & p_{2}(iz) &= -p_{2}(z), & p_{3}(iz) &= -ip_{3}(z) \\
p_{0}(-iz) &= p_{0}(z), & p_{1}(-iz) &= -ip_{1}(z), & p_{2}(-iz) &= -p_{2}(z), & p_{3}(-iz) &= ip_{3}(z)
\end{align*}
Note that
\begin{equation*}
f_{0}(z) f_{2}(z) = 1
\end{equation*}
This equation is equivalent to the following two quadratic equations:
\begin{align*}
p_{0}^{2} + p_{2}^{2} - 2p_{1} p_{3} &= 1, & p_{1}^{2} + p_{3}^{2} - 2p_{0} p_{2} &= 0
\end{align*}
Thus, you can interpret the four \(p\)-functions as coordinates in a 4-dimensional space describing certain quadratic curves.
What about differentiation? Using the infinite series definition gives
\begin{align*}
\frac{d}{dz} p_{3}(z) &= p_{2}(z), & \frac{d}{dz} p_{2}(z) &= p_{1}(z), & \frac{d}{dz} p_{1}(z) &= p_{0}(z), & \frac{d}{dz} p_{0}(z) &= p_{3}(z)
\end{align*}
This is similar to what we found for the cubic functions.
Here are some addition identities:
\begin{align*}
p_{0}(x + y) &= p_{0}(x) p_{0}(y) + p_{1}(x) p_{3}(y) + p_{2}(x) p_{2}(y) + p_{3}(x) p_{1}(y) \\
p_{1}(x + y) &= p_{0}(x) p_{1}(y) + p_{1}(x) p_{0}(y) + p_{2}(x) p_{3}(y) + p_{3}(x) p_{2}(y) \\
p_{2}(x + y) &= p_{0}(x) p_{2}(y) + p_{1}(x) p_{1}(y) + p_{2}(x) p_{0}(y) + p_{3}(x) p_{3}(y) \\
p_{3}(x + y) &= p_{0}(x) p_{3}(y) + p_{1}(x) p_{2}(y) + p_{2}(x) p_{1}(y) + p_{3}(x) p_{0}(y)
\end{align*}
Here are some doubling identities:
\begin{align*}
p_{0}(2z) &= p_{0}^{2}(z) + 2p_{1}(z) p_{3}(z) + p_{2}^{2}(z) \\
p_{1}(2z) &= 2p_{0}(z) p_{1}(z) + 2p_{2}(z) p_{3}(z) \\
p_{2}(2z) &= 2p_{0}(z) p_{2}(z) + p_{1}^{2}(z) + p_{3}^{2}(z) \\
p_{3}(2z) &= 2p_{0}(z) p_{3}(z) + 2p_{1}(z) p_{2}(z)
\end{align*}
Here are some subtraction identities:
\begin{align*}
p_{0}(x - y) &= p_{0}(x) p_{0}(y) - p_{1}(x) p_{3}(y) + p_{2}(x) p_{2}(y) - p_{3}(x) p_{1}(y) \\
p_{1}(x - y) &= -p_{0}(x) p_{1}(y) + p_{1}(x) p_{0}(y) - p_{2}(x) p_{3}(y) + p_{3}(x) p_{2}(y) \\
p_{2}(x - y) &= p_{0}(x) p_{2}(y) - p_{1}(x) p_{1}(y) + p_{2}(x) p_{0}(y) - p_{3}(x) p_{3}(y) \\
p_{3}(x - y) &= -p_{0}(x) p_{3}(y) + p_{1}(x) p_{2}(y) - p_{2}(x) p_{1}(y) + p_{3}(x) p_{0}(y)
\end{align*}
Here are some imaginary addition identities:
\begin{align*}
p_{0}(x + iy) &= p_{0}(x) p_{0}(y) -i p_{1}(x) p_{3}(y) + p_{2}(x) p_{2}(y) +i p_{3}(x) p_{1}(y) \\
p_{1}(x + iy) &= i p_{0}(x) p_{1}(y) + p_{1}(x) p_{0}(y) -i p_{2}(x) p_{3}(y) - p_{3}(x) p_{2}(y) \\
p_{2}(x + iy) &= -p_{0}(x) p_{2}(y) +i p_{1}(x) p_{1}(y) + p_{2}(x) p_{0}(y) -i p_{3}(x) p_{3}(y) \\
p_{3}(x + iy) &= -ip_{0}(x) p_{3}(y) - p_{1}(x) p_{2}(y) + ip_{2}(x) p_{1}(y) + p_{3}(x) p_{0}(y)
\end{align*}
Finally, here are some important identities:
\begin{align*}
p_{0}(x + y) + p_{0}(x - y) &= 2 p_{0}(x) p_{0}(y) + 2 p_{2}(x) p_{2}(y) \\
p_{1}(x + y) + p_{1}(x - y) &= 2 p_{1}(x) p_{0}(y) + 2 p_{3}(x) p_{2}(y) \\
p_{2}(x + y) + p_{2}(x - y) &= 2 p_{0}(x) p_{2}(y) + 2 p_{2}(x) p_{0}(y) \\
p_{3}(x + y) + p_{3}(x - y) &= 2 p_{1}(x) p_{2}(y) + 2 p_{3}(x) p_{0}(y) \\
p_{0}(x + y) - p_{0}(x - y) &= 2 p_{1}(x) p_{3}(y) + 2 p_{3}(x) p_{1}(y) \\
p_{1}(x + y) - p_{1}(x - y) &= 2 p_{0}(x) p_{1}(y) + 2 p_{2}(x) p_{3}(y) \\
p_{2}(x + y) - p_{2}(x - y) &= 2 p_{1}(x) p_{1}(y) + 2 p_{3}(x) p_{3}(y) \\
p_{3}(x + y) - p_{3}(x - y) &= 2 p_{0}(x) p_{3}(y) + 2 p_{2}(x) p_{1}(y)
\end{align*}
These will make sense later.