A Probe into the Mapping Theorem and Its Restricted Conditions and Stability Criterion

The structure of the common mapping curve is established and analyzed. The mapping theorem can be obtained plainly by starting from the concept of the loop-line,followed by linking with the zeroes and poles of complex function. Based on the numeral and interactive relationship followed by zeroes, poles and the loop-line, the interactive conditions of the formation of the mapping theorem is determined, by which the uncertainty in understanding the conclusion of N=Z-P can be avoided. In the case of the clear existence of zeroes shown by the mapping curve, the procedures of judging whether the "Z" is existent or not by N=-P can also be saved. When the number of curve anti-clockwise loops is more than given poles, the title itself should be wrong According to the restricted conditions.