4

$(1.)$

$$\sum_{n=\infty}^{\infty}f(n)$$

$(2.)$

$$\sum_{n=1}^{\infty}f(n)=\lim_{n\to\infty}\sum_{n=1}^{n}f(n)$$

How would via the tools of Complex Analysis approach the summation of series in the from defined in $(1.)$ via the tools of Complex Variables ? If possible provide applicable examples.

$$EDIT$$

Adding to our original question how would handle the upper bound and lower bound of sum defined in $(1.)$ as $n \, \rightarrow \infty$, how would generalize the approach seen in real-variable methods as seen and defined in $(2.)$.

Zophikel
  • 1,061
  • Is there a particular $f(n)$ you're interested in? – Larry B. May 16 '17 at 17:35
  • $f(n)$ can be any function really I just what to see some approach's and tools from complex variables being used on infinite sums. – Zophikel May 16 '17 at 17:37
  • 1
    Shouldn't the lower limit of the sum be $-\infty$ ? – Vivek Kaushik May 16 '17 at 17:49
  • There is Cauchy's Residue Theorem, Fourier Series with Complex Exponentials, Parseval's(Plancherel's) Identity, and Poisson Summation Formula. – Vivek Kaushik May 16 '17 at 17:50
  • @VivekKaushik interesting I should have addressed this in my question, editing now – Zophikel May 16 '17 at 19:30
  • 1
    A common method is to use that the function $g(z) = \pi\cot(\pi z)$ has residue $1$ at the integers so the integral $\oint f(z)g(z){\rm d}z$ over some appropriate contour containing all the integers will via the residue theorem give rise to $\sum f(n)$ plus additional terms coming from the poles of $f(z)$. – Winther May 17 '17 at 01:11

1 Answers1

3

I don't know what $f(n)$ you are interested in. There is Cauchy's Residue Theorem, Fourier Series with Complex Exponentials, Parseval's (Plancherel's) Identity, and Poisson Summation Formula. Have a look at the answers to proving $\zeta(2)=\frac{\pi^2}{6}$ in each of the links below: Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$) for the Residue Theorem, http://math.cmu.edu/~bwsulliv/basel-problem.pdf for Fourier Series and Parseval's Identity, http://www.libragold.com/blog/2014/12/poisson-summation-formula-and-basel-problem/ for Poisson Summation.

Vivek Kaushik
  • 3,394