|(B*z+A)/(az+b)|

=|Z|*|(B*z+A)/(az+b)|

=|(B+AZ)/(az+b)|

because

|(B+AZ) and (az+b) are replaced by * * * yoke complex numbers.

therefore

|B+AZ|=|az+b|

Goode

|(B+AZ)/(az+b)|= 1

So this equation has been proved.

Development history of complex variable function

1, the theory of complex variable function came into being in18th century. 1774, Euler considered two equations derived from the integration of complex variables in one of his papers. Before him, French mathematician D'Alembert had obtained them in his paper on fluid mechanics. Therefore, people later mentioned these two equations and called them "D'Alembert-Euler equations".

2. In the19th century, Cauchy and Riemann studied the above two equations in more detail, so they are also called Cauchy-Riemann conditions.

3. Euler and D'Alembert did the earliest work for the establishment of the theory of complex variable function, and French Laplace later studied the integration of complex variable function. They are all pioneers in establishing this discipline. Later, German mathematicians Cauchy and Riemann made peace and did a lot of basic work for the development of this discipline.

4. At the beginning of 20th century, the theory of complex variable function has made great progress. Students of Wilstrass, Swedish mathematician Levreux, French mathematicians Poincare and Adama all did a lot of research work, which opened up a broader research field of complex variable function theory and made contributions to the development of this discipline.