- #1

Ad VanderVen

- 169

- 13

- TL;DR Summary
- Simplifying a double summation.

Is it possible to simplify the function below so that the sums disappear.

$$\displaystyle g \left(x \right) \, = \, \sum _{j=-\infty}^{\infty} \left(-A +B \right) \sum _{k=-\infty}^{\infty} \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -k \right)^{2}}{\sigma ^{2}}}~\left(U -V \right)}{\sigma ~\sqrt{\pi }}$$

with

$$\displaystyle A\, = \,1/2\,{\rm erf} \left(1/2\,{\frac { \sqrt{2} \left( -j-1/2+{\it omicron} \right) }{\rho}}\right),$$

$$\displaystyle B\, = \,1/2\,{\rm erf} \left(1/2\,{\frac { \sqrt{2} \left( -j+1/2+{\it omicron} \right) }{\rho}}\right),$$

$$\displaystyle U\, = \,1/2\,{\rm erf} \left(1/4\,{\frac { \sqrt{2} \left( -2\,bj+2\,k+1 \right) }{\tau}}\right)$$

and

$$\displaystyle V\, = \,1/2\,{\rm erf} \left(1/4\,{\frac { \sqrt{2} \left( -2\,bj+2\,k-1 \right) }{\tau}}\right)$$

$$\displaystyle g \left(x \right) \, = \, \sum _{j=-\infty}^{\infty} \left(-A +B \right) \sum _{k=-\infty}^{\infty} \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -k \right)^{2}}{\sigma ^{2}}}~\left(U -V \right)}{\sigma ~\sqrt{\pi }}$$

with

$$\displaystyle A\, = \,1/2\,{\rm erf} \left(1/2\,{\frac { \sqrt{2} \left( -j-1/2+{\it omicron} \right) }{\rho}}\right),$$

$$\displaystyle B\, = \,1/2\,{\rm erf} \left(1/2\,{\frac { \sqrt{2} \left( -j+1/2+{\it omicron} \right) }{\rho}}\right),$$

$$\displaystyle U\, = \,1/2\,{\rm erf} \left(1/4\,{\frac { \sqrt{2} \left( -2\,bj+2\,k+1 \right) }{\tau}}\right)$$

and

$$\displaystyle V\, = \,1/2\,{\rm erf} \left(1/4\,{\frac { \sqrt{2} \left( -2\,bj+2\,k-1 \right) }{\tau}}\right)$$

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