# Condition in sum() function?

Hello,

I'm trying to use the exponential form of the Fourier series representation of a function to plot an approximation of said function using the first five terms.

The actual function is f(t) = 1/2 + j/(2pi) * E from (n = -5) to (n = 5) of ((-1)^n - 1)/n * e^(j2npi*t), where n =/= 0

Here I'm using 'E' to indicate summation notation. I apologize if this deviates from an established standard, but I'm having trouble uploading images right now (which would have made the function clearer).

The code I'm using for this function is

var('n, t')
j = i    # imaginary unit
expr = ((-1)^n - 1)/n * exp(2*pi*n*j*t)    # I think the problem is the division by 'n' here
assume(n, 'integer'); assume(n != 0)
f(t) = 1/2 + j/(2*pi) * sum(expr, n, -5, 5)


However, when I try this, I get the following exception:

 RuntimeError: ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative exponent. 

This seems to be due to the division by n in expr. Are my assumptions not working here? Thus, my actual question is how can I add the condition n != 0 to the sum() function in sage?

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Unfortunately, Maxima's assumption framework is not particularly strong. But Sage definitely won't know about it in your sum, since you are now using n also as a kind of dummy variable. I think it might be easier to just make two sums in this case, because 0 is only number you want to avoid in your general case, right?

( 2014-11-22 20:33:30 -0500 )edit

Yes, I only want to avoid n = 0. Thank you for your input, I ended up using the answer below.

( 2014-11-23 19:29:57 -0500 )edit

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Only t is a symbol, so you can define expr as a (Python) function of the (Python) variable n and then sum over the good set of n's.

sage: t = var('t')
sage: j = i
sage: expr = lambda n : ((-1)^n - 1)/n * exp(2*pi*n*j*t)
sage: f(t) = 1/2 + j/(2*pi) * sum([expr(n) for n in range(-5,6) if n != 0])
sage: f(t)
-1/15*I*(3*e^(10*I*pi*t) + 5*e^(6*I*pi*t) + 15*e^(2*I*pi*t) - 15*e^(-2*I*pi*t) - 5*e^(-6*I*pi*t) - 3*e^(-10*I*pi*t))/pi + 1/2

more

This is precisely what I was looking for. I didn't even think to use n as a Python variable. Thank you sir.

( 2014-11-23 19:34:40 -0500 )edit