Trouble isolating a variable in a simple equation. I am trying in vain to isolate $t$ in the equation $(t + \tau)^\alpha - t^\alpha = \beta$ with $t > 0$ and $\alpha$, $\beta$ and $\tau$ all real. Since I didn't get any useful result, I solve the equation numerically for now.
Could anyone give me a hint ? Thank you.
 A: If you are trying to solve this analytically for $t$ that's impossible. If $\alpha=n$ is an integer this is a polynomial equation of order $n-1$, and if $\alpha$ is a fraction it can be reduced to one of some degree. But polynomial equations of degree higher than $4$ can not be solved analytically according to the Galois theory.
Alternative approach is to look for positive zeros of the function $f(t):=(t + \tau)^\alpha - t^\alpha - \beta$. The derivative $f'(t)=\alpha\big((t + \tau)^{\alpha-1} - t^{\alpha-1}\big)$ is strictly negative for $t>0$ if $0<\alpha<1$, $\tau>0$, so the function is strictly monotone decreasing and there is at most one positive zero. The second derivative $f''(t)=\alpha(\alpha-1)\big((t + \tau)^{\alpha-2} - t^{\alpha-2}\big)$ is strictly positive under the same conditions, so the function is convex down. For finding zeros of strictly monotone convex functions Newton's method is very effective. It produces a sequence converging fast to the zero with any close enough initial guess. This sequence is monotone, so the error can be well estimated.
