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I just used an online calculator to calculate the following values of t.

values of t

1.) I am confused about how I am supposed to interpret the $W_{-1}$ and $W_0$. Could anyone help me out?

Here is my original equation:

$$1000e^{0.5t}=30000t$$

2.) How should I go about solving for t? I don't know Lambert W.

IV_
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Benjamin Little
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    It's hard to explain unless you know anything about complex analysis and branch cuts, in all honesty. – PrincessEev Apr 12 '22 at 05:06
  • They are the first and second branches of Lambert function (google for it) – Claude Leibovici Apr 12 '22 at 05:09
  • Alright, so how should I go about solving for t? – Benjamin Little Apr 12 '22 at 05:14
  • Your equation does not have a solution that can be expressed in terms of the usual "school level" formulas. Your past experience in school level math should be that this is apparently nearly always the case when an equation involves exponential functions and/or trig. functions, along with polynomial functions. When asking a question such as you've asked in a forum with school students to research mathematicians, it would help to give more context. For example, would a numerical approximation be enough, or do you need an explicit expression? Did this arise in school math or more advanced? – Dave L. Renfro Apr 12 '22 at 07:18

1 Answers1

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1.)

Lambert W isn't an elementary function. It's also not a function in the complex numbers. It's a relation in the complex numbers that consists of several function branches. Each of its branches is a function in the complex numbers.

The branches $W_{-1}$ and $W_0$ have real values for some of their real arguments. See the graphs of the real parts of $W_{-1}$ and $W_0$ at Wikipedia: Lambert W function.

If you need numerical values of LambertW, you need a calculator or software that has its series presentation implemented - like for all other functions and relations. Alternatively, you could use the equation $xe^x=-\frac{1}{60}$ and approximate the two x-values that satisfy this equation.

Since your online calculator found the solutions for $t$ in terms of Lambert W, it's also able to calculate the numerical values of these solutions.
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2.) Let's now calculate your solutions.

$$1000e^{0.5t}=30000t$$

The equation is an algebraic equation in dependence of the two monomials $t$ and $e^{0.5t}$. This kind of equations don't have solutions except $0$ that are elementary numbers.

Lambert W is the inverse relation of the function $x\mapsto xe^x$. We can calculate the inverse relation if we solve the equation $xe^x=y$ for $x$. We then have $x=\textrm{W}_k(y)\ \ (k\in\mathbb{Z})$. $k$ denotes the individual branches of Lambert W. To solve equations of your kind, one has to rearrange the equation into the form $f(x)e^{f(x)}=c$, where $c$ is a constant, and apply Lambert W to the left-hand side and to the right-hand side of the equation.

$$1000e^{0.5t}=30000t$$

$$e^{0.5t}=30t$$

$$1=30te^{-0.5t}$$

$$30te^{-0.5t}=1$$

$$-0.5te^{-0.5t}=-\frac{1}{60}$$

$$-0.5t=\textrm{W}_k(-\frac{1}{60}),\ k\in\mathbb{Z}$$

$$t=-\frac{\textrm{W}_k(-\frac{1}{60})}{0.5},\ k\in\mathbb{Z}$$

For real values, we need $k=-1,0$.

IV_
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  • Is there a interpretation of $\text W_n(x)$ in physics or in math besides stating its definition as an inverse function of $xe^ x$? – Тyma Gaidash Jun 04 '22 at 16:43