# Approximate the sum of exponential functions

I'm trying to approximate the sum of $$N$$ exponential functions as one function.

Is it possible to do so, i.e. find function $$f$$ and constants $$a$$, $$b$$, and $$c$$ such that $$\sum_{k=1}^N c_ke^{a_kx}\approx c f(ax+b)$$

• This is too vague. Approximate in what region? $A,B,C$ really big positive numbers? Really big negative numbers? Near $0$? And approximate how? You want the difference to go to $0$ in some limit? The ratio go to $1$? Something else?
– lulu
Jan 27 at 21:36
• Note that if you take $B=-A$ and $C=0$ we have $A+B+C=0$ so your function would need to be near (in some sense) $kf(0)$ for all such triples. But that covers a huge range of values...
– lulu
Jan 27 at 21:38
• Notice when $(A,B,C) \mapsto (A+\lambda,B+\lambda,C+\lambda)$, LHS $\mapsto e^\lambda$ LHS. If you want to preserve this basic property of exponential function, you have not much choice but setting $f(t)$ to $e^{\frac{t}{3}}$. Jan 27 at 21:42
• @lulu , A,B,C are not very big, I don't care about the way for approximation, I want the difference between LHS and RHS be close to $0$. Jan 27 at 22:06
• @achillehui , $f$ can be another function, not just exponent. Jan 27 at 22:06

It is unlikely that you will be able to approximate it if you look at it as a sum of the exponential function. Replace $$x=\ln(y)$$ and have an exponential polynomial instead.

$$\sum_{k=1}^N c_ky^{a_k}$$

Since you want an approximation, you can find the reasonable fractional approximation for all $$c_k \approx \frac{r_q}{t_q}$$ and $$a_k \approx \frac{p_k}{q_k}$$. Assume that the least common multiple of all $$q_k$$ is $$s$$, it would be then $$a_k=\frac{m_k}{s}$$. Do the same for $$t_q$$ and find that the common multiple is w making $$c_k=\frac{u_k}{w}$$

Now replace

$$z=y^{\frac1{s}}$$

$$\frac{1}{w}\sum_{k=1}^N u_kz^{m_k}$$

And $$P(z)=\sum_{k=1}^N u_kz^{m_k}$$ is a plain integer polynomial.

Other adopted values depend on the precision you want or need.