Alternative form of this equation without the product symbol?

Question

For a programming case, I need to redefine this equation, but for any value of 'n' and without the 'product' symbol

$$\prod_{i=1}^n \Bigg(1-exp\bigg(-0.5\Big(\frac{c}{s_{i}}\Big)^2\bigg)\Bigg)^{m_{i}}$$

Or at least prove that it is not possible

Thanks!

EDIT: I need all of the Si to be together in the same term, either as a sum or as a product. I do not know if it is possible though.

Answer 1

Assuming there is no equal sign, call the quantity you want $$X=\prod_{i=1}^n \Bigg(1-exp\bigg(-0.5\Big(\frac{c}{s_{i}}\Big)^2\bigg)\Bigg)^{m_{i}}$$.

Then $\log X = \sum_{i=1}^n m_i \log(1-exp\bigg(-0.5\Big(\frac{c}{s_{i}}\Big)^2\bigg)\Bigg)$. Thus $X = exp\Bigg[ \sum_{i=1}^n m_i \log(1-exp\bigg(-0.5\Big(\frac{c}{s_{i}}\Big)^2\bigg)\Bigg)\Bigg]$. Perhaps this idea may help? I'm not sure if a sum is better than a product, or what exactly your goal is here.