August 1999
I describe application from relation of integral to Taylor Series.
You may think this is not application, however. I had some doubt by chance from relation and I arrived finally here.
Although I proved, let me omit insertion of the proof.
Assuming that 
is a function defined on the 
plane, I assume also that the function can be expressed with Taylor Series as follows.
At this time, what is the necessary and sufficient condition for the intervals of 
of the degree as expressed below ?
I generalize the interpretation of the variable of from real number 
to complex number 
.
I convert into the polar form and set 
to obtain 
.
The necessary and sufficient condition for intervals of of the degree of 
is that 
satisfies constantly the formula 
.
Consequently, if 
is not a multiple of , 
, and for a derivative of order of , 
What is the value of 
at 
?
Formerly, the result could not be found without calculation. Now, the result may be obtained clearly based on the periodicity of function.
We can see that there is rule in differential coefficient.
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