Visual Method of secondary analysis: Notes: Analysis Expansion

To get a function to parse and expand a region, one way is to map a region's boundary.

Step 1:

In the above formula, the left side of the equation is the newly constructed resolution function, and its definition field p * is the reflection of the original resolution function's definition field P on the real axis. We first obtain z * from the number z in p *, and z * must be in P. Then f (z *) must be in Q. Therefore, f * (z) obtained after f (z *) is in Q.

Of course, even if there is an intersection between P and p * in the above text, we cannot say that f * (z) is an extension of f (z; because in the intersection of P and p *, f (z) and f * (z) may not be equal. However, in certain circumstances, f * (z) can indeed become the resolution extension of f (z. Let's continue.

Step 2:

In the above case, the solid axis can be promoted to a general straight line --

Step 3:

What if F is not ing a straight line to a straight line, but ing a circle to a real axis?

Step 4:

In the above formula, the function f + (z) to be mapped is defined outside the circumference, so 1/Z is in the circumference. Assume that F (1/z) maps 1/Z to the upper half plane, then f (1/z)'s [F (1/z)] * is in the lower half plane. In this way, Z and 1/Z are symmetric about the circumference, while f (z) and F + (z) are symmetric about the real.

Step 5:

"Z cross K reflection" refers to Z's symmetry points about K.