plouffe

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I wanted to see what LErchPhi function has regarding this sum for a good reason,

 

 The sum itself (without the 10^n) is sum(1/(4*n^2-4*n+4*100000000^2+1),n=1..infinity).

Looks ordinary but it is not , it is Pi , not exactly Pi for real but for all practical purpose it is

because the real answer is A * B * tanh(Pi*c) , here is an example but lower precision

sum(1/(4*n^2-4*n+4*100^2+1),n=1..infinity) = 1/800*Pi*tanh(100*Pi), if we correct the 800 term

we have then Pi*tanh(100*Pi) and this is where it is funny because this number is very close

to Pi up to the 272'th digits. So if we increase the 1000...0000 factor we can get ANY precision.

Normaly that kind of infinite sum has to do with the Psi function BUT when the arguments

are complex the 2 complex values collide and gives something completely unexpeected.

 

You see the point ?  Unfortunately by adding 10^n in the process mixes this singularity.

Too bad. In general the LerchPhi function is <unknown> for specific arguments, there is

a lengthly litterature on it BUT very few examples of explicit values.  LerchPhi(z, a, 1) is rather

simple when the argument is 1 but not when it is something else.

Best regards,

 Simon Plouffe

 

Thank you for many comments about this,

SImon Plouffe

Ok thank you but :

if you type (which is what I tried first ),

sum(1/(4*n^2-4*n+4*100000000^2+1)/10^n,n=1..infinity);

Maple returns correctly :

-1/8000000000*I*LerchPhi(1/10,1,1/2-100000000*I)+1/8000000000*I*LerchPhi(1/10,1,1/2+100000000*I)

 evalf(%);

THEN, if you want to evaluate in floating point : it <more or less> enters a trance, ... takes forever ...

So ? the point of enry make that Maple reacts differently ? this is where I don't get it.

If you just type evalf(%,12); it hangs (i. e. takes forever).

I tried of course to change the entry by adding < .0 > to all integers in the sum : did not change anything,

usually that trick works well.

but not here :

-1/8000000000*I*LerchPhi(0.1,1.0,0.5-100000000*I)+1/8000000000*I*LerchPhi(0.1,1.0,0.5+100000000*I);

takes very long to answer even if Digits: =12;

Am I missing something ?

Simon PLOUFFE

I am not certain of what you are trying to do but here is something you can use ; 

di := proc(s)
local un, de, j, p, q;
    p := numer(s);
    q := denom(s);
    un := -gamma - 1/2*Pi*cot(p*Pi/q) - ln(q);
    de := sum(cos(2*Pi*j*p/q)*ln(2*sin(j*Pi/q)), j = 1 .. q - 1);
    RETURN(un + de)
end proc

> di(1/8);
                                                    Pi                1/2           Pi       1/2          3 Pi
                               -gamma - 1/2 Pi cot(----) - 4 ln(2) + 2    ln(2 sin(----)) - 2    ln(2 sin(----))
                                                    8                               8                      8

> convert(%,radical);
                                                 1/2
                                        Pi (1 + 2   )              1/2          1/2 1/2     1/2          1/2 1/2
                               -gamma - ------------- - 4 ln(2) + 2    ln((2 - 2   )   ) - 2    ln((2 + 2   )   )
                                              2
lprint(%);
-gamma-1/2*Pi*(1+2^(1/2))-4*ln(2)+1/2*2^(1/2)*ln(2-2^(1/2))-1/2*2^(1/2)*ln(2+2^(1/2))

this formula <di> is for digamma, it is the GAUSS formula, very useful.

simon plouffe

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