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+--- Thread: [BUG] weird Zeta function (/thread-422.html)
[BUG] weird Zeta function - Tugdual - 01-12-2014 10:38 PM
Zeta(3) returns... Zeta(3). I tried Eval, no luck.
So I decided to define F1(X)=Zeta(X) but it doesn't help since F1(3) also returns Zeta(3).
I plotted F1 and then I see the curve on screen. Using the touch screen, menu, I tried "go to" and "3" "Enter". Now I got an error message: function is not defined for x=3.
I tried a manual approximation and entered sum(n^ -3,n,1,1000) which returns 1.20 which is close enough and consistent with what I see on the curve.
So why zeta(3) fails?
Also the zeta help says that Re(x) shall be >1 while zeta returns values for negative x?
RE: weird Zeta function - Mark Hardman - 01-12-2014 10:55 PM
(01-12-2014 10:38 PM)Tugdual Wrote: Zeta(3) returns... Zeta(3).
In CAS mode, forcing the argument to be real does provide an approximation of Apéry's constant.
Zeta(3.) = 1.20205690316
But this doesn't work in Home mode.
Mark Hardman
RE: weird Zeta function - Tugdual - 01-13-2014 06:55 AM
(01-12-2014 10:55 PM)Mark Hardman Wrote: In CAS mode, forcing the argument to be real does provide an approximation of Apéry's constant.
Zeta(3.) = 1.20205690316
Hi Mark, thanks for the hint!
I find it even more bizarre that the only way to get an approximation is in CAS which seems to be dedicated to symbolic calculations.
Also in the meantime I tried Zeta(1+i) it says Eta(1+i)/(1-2^-i)
I think this is correct but suprisingly we see Dirichelt's Eta function popping up while it is neither documented nor listed in the catalog.
Note: I have edited the first post to call it a BUG
RE: [BUG] weird Zeta function - Helge Gabert - 01-13-2014 04:58 PM
Yes,
This was already noted a while back (if you peruse the archives).
Dirichlet's eta function also won't get evaluated numerically.
RE: [BUG] weird Zeta function - Joe Horn - 01-13-2014 07:51 PM
(01-13-2014 06:55 AM)Tugdual Wrote: I find it even more bizarre that the only way to get an approximation is in CAS which seems to be dedicated to symbolic calculations.
Zeta(approx(3)) returns a real in Home. So does turning off "Change apparent integers into exact integers" in the CAS Settings (page 1, end of 3rd line), but since that's not programmable, I suggest using the Zeta(approx(3)) method.
These two methods are related. If you type Zeta(3.) in Home, CAS turns the real 3 into an integer automatically (not a bug; it's due to that CAS setting), hence the symbolic result. But approx is a CAS function, so it turns approx(3) into a real 3 in the CAS, and feeds that to the Zeta function. OR you can turn off the "convert whole numbers into integers" mode, and avoid using approx.
RE: [BUG] weird Zeta function - Tugdual - 01-13-2014 09:48 PM
(01-13-2014 07:51 PM)Joe Horn Wrote:Hi Joe, that's interesting, I didn't even notice this option in CAS parameters (a simple check box with no further explanation...). I understand what you say but I'm happy I don't have to enter sin(approx(3)) everytime I need to do a sinus calculation ;-)(01-13-2014 06:55 AM)Tugdual Wrote: I find it even more bizarre that the only way to get an approximation is in CAS which seems to be dedicated to symbolic calculations.
Zeta(approx(3)) returns a real in Home. So does turning off "Change apparent integers into exact integers" in the CAS Settings (page 1, end of 3rd line), but since that's not programmable, I suggest using the Zeta(approx(3)) method.
These two methods are related. If you type Zeta(3.) in Home, CAS turns the real 3 into an integer automatically (not a bug; it's due to that CAS setting), hence the symbolic result. But approx is a CAS function, so it turns approx(3) into a real 3 in the CAS, and feeds that to the Zeta function. OR you can turn off the "convert whole numbers into integers" mode, and avoid using approx.
RE: [BUG] weird Zeta function - Joe Horn - 01-16-2014 07:23 PM
(01-13-2014 09:48 PM)Tugdual Wrote:The explanation is at the bottom of the screen, immediately above the soft keys. Those prompts are often the only explanation offered by input forms.(01-13-2014 07:51 PM)Joe Horn Wrote: ... "Change apparent integers into exact integers" in the CAS Settings (page 1, end of 3rd line) ...... I didn't even notice this option in CAS parameters (a simple check box with no further explanation...)
RE: [BUG] weird Zeta function - Curlytop - 01-17-2014 09:33 PM
I had noticed this phenomenon with the Prime. My own opinion FWIW is that Zeta(r) is evaluated in terms of Zeta(r-2), recursively until a value between 0 and 2 is reached. Faced with Zeta() of an odd integer, it will backtrack successfully until it reaches Zeta(1) and boggles.
RE: [BUG] weird Zeta function - parisse - 01-18-2014 07:17 AM
Zeta(n) is left as is if n>0 is odd, and computed using bernoulli numbers if n is even. You can't do any better if you want an exact answer. Of course, you can enter Zeta(d) where d is a floating point number to get an approximation.
Code:
...
if ( (x.type==_INT_)){
int n=x.val;
if (!n)
return minus_one_half;
if (n==1)
return plus_inf;
if (n<0){
if (n%2)
return -rdiv(bernoulli(1-n),(1-n),contextptr) ;
else
return zero;
}
if (n%2)
return symbolic(at_Zeta,x);
else
return pow(cst_pi,n)*ratnormal(abs(bernoulli(x),contextptr)*rdiv(pow(plus_two,n-1),factorial(n),contextptr));
}
...RE: [BUG] weird Zeta function - Curlytop - 01-18-2014 11:01 AM
Yet if I ask for Zeta(3) on my WP-34S, I get the perfectly sensible answer of 1.20205690316, the same answer that Zeta(approx(3)) gives on the Prime.
RE: [BUG] weird Zeta function - Tugdual - 01-19-2014 01:16 PM
So on top of hidden Eta (see my previous posts) I see that we also have Benouilli numbers. Why don't we have access to those functions?
Also I agree with Curlytop, why don't you simply cast to real and return an estimate instead of "symbolic(at_Zeta,x);"?