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Question: How to get a general expression for derative ?

Question:How to get a general expression for derative ?

janhardo 555 Maple
complex differentiation duplicate-question
January 28 2022
1

How to get a general expression here ?

restart

 

f(s) is the function value  expressed as a integral around s  (singularity)  of a complex function.

f(s) = (int(f(z)/(z-s), z))/(2*Pi*I)

f(s) = -((1/2)*I)*(int(f(z)/(z-s), z))/Pi

 (1)
> 

f(s) = int(f(z)/(z - s), z)/((2*Pi)*I):

"(->)"

diff(f(s), s) = -((1/2)*I)*(int(f(z)/(z-s)^2, z))/Pi

 (2)

"(->)"

diff(diff(f(s), s), s) = -((1/2)*I)*(int(2*f(z)/(z-s)^3, z))/Pi

 (3)

"((ⅆ)^(n))/(ⅆn) f(s) = "
                                      

NULL

NULL

                 .......

quote : "Important consequence.

Above it actually says: "If there is a function f(s) that is somewhere analytic, then you can use such an integral as above to make a new function f '(s), which is also analytic there. And from that a new function f '' and so on.
That means something revolutionary for complex numbers:   "

 

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Question: how to get the "((ⅆ)^(n))/(ⅆn) f(s) "?
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Note : again the form of the answers in Maple : Its not possible to force Maple to come up with this form of answer, ex

``  NULL

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