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I have a problem to calculate 'k' from this equation :

$$X = \frac{\left(\rho-\rho^{k+1}\right)\left(1-\frac{\gamma}{2}\rho\right)^{2}-k\rho^{k}\left(1-\frac{\gamma}{2}\rho\right)}{\lambda\left(1-\rho\right)\left(1-\rho^{k}\right)}$$

When I expand this equation, it will be :

$\rho^{k}$(some equations) - $k\rho^{k}$(some equations) = value containing X and $\lambda$

because there is 'k' in $k\rho^{k}$, I can not calculate k from given X and $\lambda$.

Could anyone know how to find k from this equation?

Thank you

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2 Answers

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By multiplying both sides of equation by some constant, you will get the equation like $$(A+k)\rho^k=B(X,\lambda)$$ Or, $$(A+k)\rho^{A+k}=(A+k)e^{(A+k)\ln\rho}=B\rho^A\\ \left((A+k)\ln\rho\right)e^{(A+k)\ln\rho}=B\rho^A\ln\rho$$ Then you will refer to Lambert W function and obtain $$(A+k)\ln\rho=W_e(B\rho^A\ln\rho)$$ and then the solution is $$k=\frac{W_e(B\rho^A\ln\rho)}{\ln\rho}-A$$

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Let u = ($1-\gamma p/2$) and w = $\lambda$/(1-p) just so we can write things a little more simply. The equation becomes:

X = $[p(1-p^k)u^2 -ukp^k]$/($w(1-p^k)$ = $pu^2/w -(ukp^k)/[(1-p^k]$

=$(pu/w)[u -kp^{k-1}/(1-p^k)]$

$(wX/pu) -u = -kp^{k-1}/(1-p^k) = d[(1-p^k]/dk)$

$1-p^k = \int (wXu -u)dk =(wXu -u)k$

$(1-p^k)/k = (wXu -u)$

If my arithmetic is correct, this may be the best you can do.

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