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This question is motivated by the response provided in this question

Considering the same equation which is shown below $$c = 1 - \exp\left(\lambda_1 R^2 \left(\frac{2\pi}{3}-\frac{\sqrt{3}}{2}\right) + \lambda_3 R^2 \left(\frac{2\pi}{3}-\frac{\sqrt{3}}{2}\right)\right)$$

It has been proven that it cannot be expressed as a sum. Still, I am wondering if I can express this as a product like $c = x * y$, where $x$ is in terms of $\lambda_1$, and $y$ is in terms of $\lambda_3 $? I will appreciate insights on this matter.

Pardon, I am asking a similar question, but as you can see, the dimensions are different.

Abdulhameed
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1 Answers1

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Similarly as for the additive case, this is not possible unless $\lambda_1=0$ or $\lambda_3=0$ or $R=0$.

Suppose there exist two functions $x$ and $y$ such that $$x(\lambda_1)y(\lambda_3)=1 - \exp\left(\lambda_1 p + \lambda_3 q\right)$$ Then note that $x(0)y(0)=1-1=0$. Also $$ x(\lambda_1)y(0)=1-e^{\lambda_1p}$$ $$ x(0)y(\lambda_3)=1-e^{\lambda_3q}$$ Multiplying the two yields $$ x(\lambda_1)y(\lambda_3)x(0)y(0)=(1-e^{\lambda_1p})(1-e^{\lambda_3q})$$ But since $x(0)y(0)=0$, that means $$(1-e^{\lambda_1p})(1-e^{\lambda_3q})=0$$ which implies that $\lambda_1=0$ or $\lambda_3=0$ or $R=0$.

Stefan Lafon
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