Inverse of Exponential Integral funtion?

Question

So $Ei(x) =- \int_{-x}^{\infty} \frac{e^{-t}}t \mathrm dt$.

Now say I have $Ei(mx) = Ei(r) + ye^r$.

Can I write this equation with $x$ on the LHS of the equation?

Can this be written?

If so, with what equation?

Answer 1

Symbolically, the solution is given by

$$x=\frac1m\operatorname{Ei}^{-1}(\operatorname{Ei}(r)+ye^r)$$

and the main problem is calculating $\operatorname{Ei}^{-1}$. This can be done numerically using Newton's method.

Let $z=\operatorname{Ei}(r)+ye^r$ and $t=\operatorname{Ei}^{-1}(z)$. It then follows that

$$z=\operatorname{Ei}(t)\implies z=\operatorname{li}(e^t)$$

$t$ may then be calculated numerically via Halley's method, as is described in this answer.

Once you know $t$, divide by $m$ and you will have solved for $x$.

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You could also use Newton's method. Let $u=e^t$ and $u_0=z$.

$$u_{n+1}=u_n-\ln(u_n)(\operatorname{li}(u_n)-z)$$

For example, with $z=10$,

$u_0=10$

$u_1=18.8290334208$

$u_2=20.2659926495$

$u_3=20.2843626902$

$u_4=20.2843654566$

$u_5=20.2843654566$

Thus, $u\approx20.2843654566$, so $t\approx3.00985041476$