To illustrate, consider a sexp solution \( S(z) \) and its inverse \( S^{\small -1}(z) \). Let's say that, in the vicinity of the upper fixed point c_0, \( S^{\small -1}(z) \approx \log_{c_0}\left(z-c_0\right) \).
Now let's say we have an alternate solution \( R(z) \) and its inverse \( R^{\small -1}(z) \). Let's say that \( R(z)=S\left(z+\frac{\sin(2\pi z)}{200\pi}\right) \).
This shift is a pretty small one. After all:
\(
\begin{eqnarray}
S^{\small -1}\left(S(z)\right) & = & z \\
\\[5pt]
\\
D_z \left[S^{\small -1}\left(S(z)\right)\right] & = & D_z \left[z\right] \\
& = & 1 \\
\\[15pt]
\\
S^{\small -1}\left(R(z)\right) & = & z+\frac{\sin(2\pi z)}{200\pi} \\
\\[5pt]
\\
D_z \left[S^{\small -1}\left(R(z)\right)\right]
& = & D_z \left[z+\frac{\sin(2\pi z)}{200\pi}\right] \\
& = & 1 + \frac{\cos(2\pi z)}{100}
\end{eqnarray}
\)
As you can see, for real z, the shift is small, as is the derivative. In fact, for real z, the derivative oscillates between 0.99 and 1.01.
But what happens to \( S^{\small -1}\left(R(z)\right) \) as we approach z values with large imaginary part? Well, to get an idea, let's look at the derivative for a z value with a relatively small imaginary part.
\(
\begin{eqnarray}
S^{\small -1}\left(R(2+2i)\right) & = & z+\frac{\sin\left(2\pi (2 + 2i)\right)}{200\pi} \\
& = & 2 + 228.19i \\
\\[5pt]
\\
D_z \left[S^{\small -1}\left(R(2+2i)\right)\right]
& = & D_z \left[z+\frac{\sin\left(2\pi (2 + 2i)\right)}{200\pi}\right] \\
& = & 1434.76
\end{eqnarray}
\)
In fact, here's a chart to give an idea of how big these oscillations can get:
Now let's say we have an alternate solution \( R(z) \) and its inverse \( R^{\small -1}(z) \). Let's say that \( R(z)=S\left(z+\frac{\sin(2\pi z)}{200\pi}\right) \).
This shift is a pretty small one. After all:
\(
\begin{eqnarray}
S^{\small -1}\left(S(z)\right) & = & z \\
\\[5pt]
\\
D_z \left[S^{\small -1}\left(S(z)\right)\right] & = & D_z \left[z\right] \\
& = & 1 \\
\\[15pt]
\\
S^{\small -1}\left(R(z)\right) & = & z+\frac{\sin(2\pi z)}{200\pi} \\
\\[5pt]
\\
D_z \left[S^{\small -1}\left(R(z)\right)\right]
& = & D_z \left[z+\frac{\sin(2\pi z)}{200\pi}\right] \\
& = & 1 + \frac{\cos(2\pi z)}{100}
\end{eqnarray}
\)
As you can see, for real z, the shift is small, as is the derivative. In fact, for real z, the derivative oscillates between 0.99 and 1.01.
But what happens to \( S^{\small -1}\left(R(z)\right) \) as we approach z values with large imaginary part? Well, to get an idea, let's look at the derivative for a z value with a relatively small imaginary part.
\(
\begin{eqnarray}
S^{\small -1}\left(R(2+2i)\right) & = & z+\frac{\sin\left(2\pi (2 + 2i)\right)}{200\pi} \\
& = & 2 + 228.19i \\
\\[5pt]
\\
D_z \left[S^{\small -1}\left(R(2+2i)\right)\right]
& = & D_z \left[z+\frac{\sin\left(2\pi (2 + 2i)\right)}{200\pi}\right] \\
& = & 1434.76
\end{eqnarray}
\)
In fact, here's a chart to give an idea of how big these oscillations can get:
Code:
| z | S^-1(R(z)) | D_z S^-1(R(z))
+----------------+----------------------+----------------------
| 2.00 | 2.00 | 1.01
| 2.25 | 2.25 | 1.00
| 2.50 | 2.50 | 0.99
| 2.75 | 2.75 | 1.00
| 3.00 | 3.00 | 1.01
| 3.25 | 3.25 | 1.00
| 3.50 | 3.50 | 0.99
| 3.75 | 3.75 | 1.00
| 4.00 | 4.00 | 1.01
| | |
| 2.00 + 1.00 i | 2.00 + 0.43 i | 3.68
| 2.25 + 1.00 i | 2.68 | 1.00 - 2.68 i
| 2.50 + 1.00 i | 2.50 - 0.43 i | -1.68
| 2.75 + 1.00 i | 2.32 | 1.00 + 2.68 i
| 3.00 + 1.00 i | 3.00 + 0.43 i | 3.68
| 3.25 + 1.00 i | 3.68 | 1.00 - 2.68 i
| 3.50 + 1.00 i | 3.50 - 0.43 i | -1.68
| 3.75 + 1.00 i | 3.32 | 1.00 + 2.68 i
| 4.50 + 1.00 i | 4.00 + 0.43 i | 3.68
| | |
| 2.00 + 2.00 i | 2.00 + 228.19 i | 1434.76
| 2.25 + 2.00 i | 230.44 | 1.00 - 1433.76 i
| 2.50 + 2.00 i | 2.50 - 228.19 i | -1432.76
| 2.75 + 2.00 i | -225.44 | 1.00 + 1433.76 i
| 3.00 + 2.00 i | 3.00 + 228.19 i | 1434.76
| 3.25 + 2.00 i | 231.44 | 1.00 - 1433.76 i
| 3.50 + 2.00 i | 3.50 - 228.19 i | -1432.76
| 3.75 + 2.00 i | -224.44 | 1.00 + 1433.76 i
| 4.50 + 2.00 i | 4.00 + 228.19 i | 1434.76
~ Jay Daniel Fox