janhardo

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8 years, 46 days

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These are questions asked by janhardo

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:   "

 

================================

Question: how to get the "((ⅆ)^(n))/(ⅆn) f(s)  "?
===============================

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

Download Maple_primes_bvraag_hoger_orde_singulariteit_henk_hofstede.mw

Seems to me informative to see a earth-like surface on a sphere and in particular from the zeta function.
Another simple complex function will do it also.

Note: i saw a  3D picture with a colored sphere , where you can see zeroes and poles on the surface of the sphere
A colored complex function is that hard to make with Maple too?: it are all polar coordinates as complex points in the complex plane. 
The angle is standing for hue and the magnitude is standing for the lightness 

Really , the complex plotting possibilities in Maple are difficult to decipher. 

I tried something, but  in general the visualizing for me is not that easy

Now i must look at the complexplot where i got a circle for  the complexplot(sin(x + I), x = -Pi .. Pi) example ?

Try to calculate the values for the Zeta(z) on the critical line in symbolic form and also as a plot 

Both i did not yet succeed in 

f:=z->Zeta(z);

complexplot3d(f,-1-1*I..1+1*I);
 

solve(f = 1/2 , z ) 

 

Alternating serie

 

sum((-1)^(n+1)/(2*n-1), n = 1 .. infinity)

(1/4)*Pi

(1)

 

sum((-1)^(n + 1)/(2*n - 1), n = 1 .. infinity):

sum((-1)^(n+1)/(2*n-1), n = 1 .. 4)

76/105

(2)

expand(sum((-1)^(n + 1)/(2*n - 1), n = 1 .. 4),symbolic);

76/105

(3)

?expand

series(sum((-1)^(n + 1)/(2*n - 1), n = 1 .. infinity),n=0,5);

series((1/4)*Pi,n)

(4)

 

Info series

   

How to get for n= 4  "for (∑)(((-1)^(n+1))/(2 n-1))  =   symbolic term 1+ symbolic term 2+... "

 

 

Download onderzoek_reeks_-hoe_krijg_ik_een_partieke_symbolische_som.mw

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