houlahound
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Can a table of values for W be computed?
ie
z in cloum a, W(z) in colum b.
ie
z in cloum a, W(z) in colum b.
Looking for internet famous math prob on dist law.
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houlahound said:
Yes, in exactly the same way that one can be computed for the log and the sin function.
houlahound
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K, I will give the link a proper read, I stopped when it defined W as inverse branches of something else, what does that even mean.
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It means that the ##W## function is an inverse function.
For example, ##log(z) = w## if and only if ##e^w = z##. This means that the ##\log## is the inverse function of the exponential.
But situations like the log and the exponential aren't always this simple. Take ##w =\sqrt{z}##. It is certainly true that ##w^2 = z##, but ##w## isn't the only value for which ##w^2 = z##, also ##(-w)^2 = z##. So there are two solutions to ##w^2 = z##. We choose one solution (namely the positive one) and call that ##\sqrt{z}##. But we have made a quite arbitrary choice here, we could also take the negative. We say that the square root function has two branches (one for each choice). So there is a negative branch of numbers whose square is ##z## and a positive branch.
In the same way, there is no unique ##w## such that ##we^w = z##. A choice must be made (in some case, in others there is a unique solution). These two choices define two "branches" of the W function.
For example, ##log(z) = w## if and only if ##e^w = z##. This means that the ##\log## is the inverse function of the exponential.
But situations like the log and the exponential aren't always this simple. Take ##w =\sqrt{z}##. It is certainly true that ##w^2 = z##, but ##w## isn't the only value for which ##w^2 = z##, also ##(-w)^2 = z##. So there are two solutions to ##w^2 = z##. We choose one solution (namely the positive one) and call that ##\sqrt{z}##. But we have made a quite arbitrary choice here, we could also take the negative. We say that the square root function has two branches (one for each choice). So there is a negative branch of numbers whose square is ##z## and a positive branch.
In the same way, there is no unique ##w## such that ##we^w = z##. A choice must be made (in some case, in others there is a unique solution). These two choices define two "branches" of the W function.
houlahound
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OK this I can do, expansion of W
This also
ETA
This also
ETA
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houlahound said:OK this I can do, expansion of W
This maybe
Yes, in the same way that the sin and the logarithm have such expansions! You see, there is little difference between the W and the logarithm. The only difference is that you've learned the log at a young age and thus are more "used" to it.
houlahound
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But this discussion started that W can not be expressed by more elementary functions when from above clearly it can.
I have really enjoyed and value this discussion cos I got some new math. Appreciate you sticking with.
I have really enjoyed and value this discussion cos I got some new math. Appreciate you sticking with.
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houlahound said:
The term elementary function is a technical term with a very specific meaning. You shouldn't put too much importance at these terms. A function being not-elementary doesn't mean it's hard to compute or anything.