August 1999

Application to Taylor Series

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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