M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

Quartic Functions 1


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.