definition
hypothesis function and in Area in Analysis, has a common part with to be established, so and See the numerical collection on all points in and , in < In / Section> Analysis, in , in .
function You can see the defined area of expansion , which is called the resolution of . Of course, according to the same reason, is parsing, this way to expand the original to function, referred to as parsing extender .
wants to make this method meaning, must only give only the only result in the appropriate conditions, we will prove that it is true, before giveing it, first mention, if A similar method is defined for the unit real function, what difficult will be encountered.
is set in , , people will naturally recommend using this formula to expand the definition of to other < / section>. However, difficulties are that two different formulas may be the same function in a certain range, but in another interval, different functions, and there is no obvious reason to determine which formula is "just". For example, in , the above function can also be used as a magnitude
is equal to .
This level is inconsistent convergence, but even if the same thing is unanimously converge, the same thing will happen, such as the number of levels
in one included < Section>interval unanimously converge; if it uses it, it is not the hope that the number of the number of and from the positive is not hoped, Exvelor of Section> is .
As a general parsing function definition
parsing function is often defined in a certain restricted area of the plane, the principles of extending make us define a definition there is Any parsing function of any special area, which consists of all the extension of the original function and the original function and all of these extendroans. This way we may define functions in all , or make it definition on any point outside some special points, or only a certain limit on the plane. There is definition in the area, but no longer overcome it. In the final situation, the region is the existence area of the function, and its boundary is called the natural boundary of the function. For the case of multi-value functions, we will get multiple different values of the function on some z or on all Zs. .
It seems that this definition depends on the special definition of the functions we started, but the relationship between the two mutual extension is reversible, so all the processes can be reversed. Come over, so this definition is actually not related to any special starting point.
Dandu standard method
extending standard method is the power level method, assuming we start from the number of grades
It converges in circular . Take a point B in the circle, calculate the value of and the value of the dividers Mode. This level must converge in any circle of B as a center and completely fall in the original circle, which may also converge in a larger circle, providing a function of a resolution of the function, so the entire function can Configuration through the power level. Each power level, or each set of value of it is called a function.
The following theorem illustrates the reason why the value obtained by any extendroid method can also be obtained by a power-grade method.
You C is a connection point z = a and Z = b, along this circumference, we have used some way to extend the , There is a formula that defines in the area , and has the nature of the secondary: (i) each point of C is one Or multiple D n of the inner point; (ii) successive overlap each other, and on the public part, the different definitions of have the same Value.
We must use the power-level method to achieve this, that is, to find a list to make the converging circle in each point in the column contain the next point, and The value obtained by the power series method is the same as the other method. In this way, it will be able to reach B after a limited multiple steps.
For each point of C, there is a positive convergence radius is a continuous function of Z, if For the neighboring two points, and the convergence radius of the "Section> and table," section> . Because is in the circle of the radius in as the radius, it is derived from the Cori Pleulele theorem to
, use the same method, but use
< Section> Because it is , it is , in any case, (2) constant. Together (1) and (2), it is proved that when is , this is the result of the need.
Because continuous, it will be able to get the lower default; because it constant, it must be a positive number, set this assure to .
We start from , order is point along the distance and A distance equal to , it Within the convergence circle at a point, the function exhibits the function exhibiting the power series of , the new convergence radius is at least , so it can reach the curve and A Point from , followed by this method, it is clear that it will arrive at B after a limited time. As for the value of B obtained by this method, the fact that is equal to other methods can be launched by a general uniqueness.
