Is there a way to simplify this expression $ (i\cdot T+ e^{-i\cdot T} -1)$

Question

I've been using Maple to solve some problems at hand. In particular, when I solve this integral

$$ \int_{0}^{T} {(e^{i\cdot (t - T)} - 1)\cdot (N-a) \over b} dt = S $$

I get the following solution:

$$ {(i\cdot T+ e^{-i\cdot T} -1)\cdot (a-N) \over b\cdot i} = S $$

So when trying to isolate the expression for $T$, I get a scary expression like in this (screenshot). As you can see, the solution involves Lambert W which I'd like to avoid (since I still need to code on GAMS and I'm clueless how to deal with this).

My question is there any way to simplify this expression $ (i.T+ \exp^{-i.T} -1) $ to avoid the Lambert W in the solution for $T$?

$T$ is not a small number, it equals 100. $i$ is the interest rate.

Answer 1

Let $$x=i\,T \qquad \text{and} \qquad k=1+\frac{b\, i\, S}{a-N}$$ and you need to solve for $x$ $$x+e^{-x}=k \tag 1$$

Beside Lambert, you could use series expansion and power series reversion to obtain $$x=y+\frac{y^2}{6}+\frac{y^3}{36}+\frac{y^4}{270}+\frac{y^5}{4320}- \frac{y^6}{17010}+O\left(y^7\right)$$ where $y= \sqrt{2(k-1)}$.

Plugging in $(1)$ $$x+e^{-x}=k+\frac{139 }{340200}(k-1)^4$$

You can also take more terms and transform the series in a corresponding Padé approximant such as $$x=y\,\frac{1-\frac{36 }{107}y+\frac{145 }{2996}y^2-\frac{37 }{77040} y^3}{ 1-\frac{323 }{642}y+\frac{313 }{2996}y^2-\frac{12329}{1617840}y^3}$$ whose error is $\frac{419 }{9784696320}y^8$

Plugging in $(1)$ $$x+e^{-x}=k-\frac{419 }{305771760 \sqrt{2}}(k-1)^{9/2}$$

Answer 2

$${(iT+e^{-iT}-1)(a-N)\over bi}=S$$ $$(iT+e^{-iT}-1)(a-N)=biS$$ $$(iT+e^{-iT}-1)(a-N)-biS=0$$

We see, this equation is a polynomial equation of more than one algebraically independent monomials ($T,e^{-iT}$) and with no univariate factor. We therefore don't know how to rearrange the equation for $T$ by applying only finite numbers of elementary functions (elementary operations) we can read from the equation.

$$(N-biS+(ai-iN)T-a)e^{iT}+a-N=0$$ $T\to\frac{t}{i}$: $$(N-a-biS+(a-N)t)e^t+a-N=0$$

In the case of algebraic $a,b$ and $S$, this equation is an algebraic equation of both $t$ and $e^t$ with no univariate factor and the equation cannot have solutions except $0$ that are elementary numbers therefore in this case.

For series representations and software algorithms for Lambert W, search Google for "On the Lambert W Function".