A Problem Suggested by

[@XJWill1]

If you would like to visualize the solutions in the complex xy-plane, you can plot the families of curves implicitly defined by the two equations below and look for the intersection points. As usual, I assumed the principal values of all multi-valued functions. It is interesting to note that the solutions all come in conjugate pairs. It is a good exercise to prove that if z = x + i * y solves the given equation, then so does z = x - i * y. Hint: it is related to the fact that i^i is real-valued. Here are the implicitly defined families of curves to plot: pi = 4*y*arctan(y / (sqrt(x^2 + y^2) + x)) - x*ln(x^2 + y^2) 0 = tan( x*arctan(y / (sqrt(x^2 + y^2) + x)) + y/4 *ln(x^2 + y^2) ) Another interesting thing to see is that there is only one pair of solutions with negative real part, but an infinite number of solutions with positive real part.

[@scottleung9587]

I almost got exactly what you got - I just missed the imaginary part in the Lambert expression.

[@SIB1963]

Lambert's W is clever, but it begs the question. If you have to use Wolfram Alpha to solve the problem, then all of the work you do to get to that answer is just setting the problem up for a computer to solve for you. How do you analytically determine W( f(z) ) for some given function f? Or alternatively, how do you analytically solve the problem without resorting to Lambert's W function?

[@XJWill1]

ln() is clever, but it begs the question. If you have to use Wolfram Alpha to solve the problem, then all of the work you do to get to that answer is just setting the problem up for a computer to solve for you. How do you analytically determine ln( f(z) ) for some given function f? Or alternatively, how do you analytically solve the problem without resorting to ln() function?

[@SIB1963]

@@XJWill1 Great comment. This really points up my deficiency, which is a lack of deep understanding of the nature of many functions. I am no mathematician. I understand how ln() is the inverse function of e exponentiation, even if I would have great difficulty working out a log by hand. Somehow, I need to view Lambert's W in the same way. It's a bit hard for me to wrap my head around. I want to understand algorithms for how to work out numbers, feeling somehow that if I know how to calculate it on paper, I in some sense "know" it better or it becomes more "real" to me or something.

[@XJWill1]

@@SIB1963 If you want to better understand the product-log function, I suggest starting with studyng graphs of the function. You can easily plot the real-valued product-log function in Desmos using x = y * exp(y) and after you have studied that (I suggest paying special attention to the domain and range, and where it is multi-valued), you might want to consider the complex-valued product-log, which is actually an infinite set of curves (similar to the complex logarithm function which is infinitely multi-valued).

[@Nobodyman181]

e^ln(i)=i

[@bobbyheffley4955]

You can obtain z=i by inspection.

[@mcwulf25]

Of course you can. But do you really think that's all there is?

[@MAZHOR1113]

я первый