Confusing identities in lambert w function that W(x)e^W(x)=x=W(xe^x), how is this possible ?<\/p>\n\n\n\n
You\u2019ve hit on the most important \u2014 and initially confusing \u2014 property of the Lambert W function. Let\u2019s break it down step by step.<\/p>\n\n\n\n
—<\/p>\n\n\n\n
1. Definition of \\( W \\)<\/p>\n\n\n\n
The Lambert \\( W \\) function is defined as the inverse of the function:
\\[
f(w) = w e^w
\\]
That means:
\\[
\\text{If } y = w e^w, \\quad \\text{then } w = W(y).
\\]<\/p>\n\n\n\n
So by definition :
\\[
W(x) e^{W(x)} = x
\\]
for any \\( x \\) where \\( W(x) \\) is defined.<\/p>\n\n\n\n
—<\/p>\n\n\n\n
2. Second identity: \\( W(x e^x) = x \\)<\/p>\n\n\n\n
Let\u2019s check this one carefully.<\/p>\n\n\n\n
We want to solve for \\( W(x e^x) \\).<\/p>\n\n\n\n
Let \\( t = x e^x \\).<\/p>\n\n\n\n
By definition of \\( W \\), \\( W(t) \\) is the number such that:
\\[
W(t) e^{W(t)} = t
\\]
But \\( t = x e^x \\), so:
\\[
W(x e^x) e^{W(x e^x)} = x e^x
\\]<\/p>\n\n\n\n
Now, notice that \\( x \\) itself satisfies \\( x e^x = t \\).
Since \\( W \\) is the inverse of \\( w \\mapsto w e^w \\), applying \\( W \\) to \\( t \\) should return \\( x \\) if \\( x \\) is in the correct domain (i.e., \\( x \\ge -1 \\) for the principal branch \\( W_0 \\)).<\/p>\n\n\n\n
So indeed:
\\[
W(x e^x) = x
\\]
for \\( x \\ge -1 \\) (principal branch).<\/p>\n\n\n\n
—<\/p>\n\n\n\n
3. Why both identities look similar but are different<\/p>\n\n\n\n
First identity: \\( W(x) e^{W(x)} = x \\)
This is the direct definition of \\( W \\): \u201c\\( W(x) \\) is the number whose product with \\( e^{W(x)} \\) equals \\( x \\).\u201d<\/p>\n\n\n\n
Second identity: \\( W(x e^x) = x \\)
This is the inverse property: \u201cIf you take \\( x \\), form \\( x e^x \\), then apply \\( W \\), you get back \\( x \\).\u201d<\/p>\n\n\n\n
—<\/p>\n\n\n\n
4. Analogy with logarithm<\/p>\n\n\n\n
Think of the natural log function \\( \\ln(x) \\):<\/p>\n\n\n\n
– Definitional property: \\( e^{\\ln x} = x \\)
– Inverse property: \\( \\ln(e^x) = x \\)<\/p>\n\n\n\n
Similarly for \\( W \\):<\/p>\n\n\n\n
– Definitional: \\( W(x) e^{W(x)} = x \\)
– Inverse: \\( W(x e^x) = x \\)<\/p>\n\n\n\n
—<\/p>\n\n\n\n
5. Example with numbers<\/p>\n\n\n\n
Let \\( x = 2 \\):<\/p>\n\n\n\n
– Compute \\( W(2 e^2) \\):
\\( 2 e^2 \\approx 14.778 \\)
\\( W(14.778) \\) is the number \\( w \\) such that \\( w e^w = 14.778 \\).
Try \\( w=2 \\): \\( 2 e^2 \\approx 14.778 \\) \u2713
So \\( W(2 e^2) = 2 \\). This illustrates \\( W(x e^x) = x \\).<\/p>\n\n\n\n
– Compute \\( W(2) \\):
\\( W(2) \\approx 0.8526 \\)
Check: \\( 0.8526 \\times e^{0.8526} \\approx 0.8526 \\times 2.345 = 2.000 \\) \u2713
This illustrates \\( W(x) e^{W(x)} = x \\).<\/p>\n\n\n\n
—<\/p>\n\n\n\n
6. Domain considerations<\/p>\n\n\n\n
– \\( W(x e^x) = x \\) holds for \\( x \\ge -1 \\) for the principal branch \\( W_0 \\), and for \\( x \\le -1 \\) for the \\( W_{-1} \\) branch.
– \\( W(x) e^{W(x)} = x \\) holds for all \\( x \\) in the domain of \\( W \\) (which is \\( x \\ge -1\/e \\) for real \\( W \\)).<\/p>\n\n\n\n
—<\/p>\n\n\n\n
7. Summary<\/p>\n\n\n\n
Both identities express that \\( W \\) and the map \\( w \\mapsto w e^w \\) are inverses:<\/p>\n\n\n\n
– \\( W(x) e^{W(x)} = x \\) says: applying \\( w e^w \\) to \\( W(x) \\) gives \\( x \\).
– \\( W(x e^x) = x \\) says: applying \\( W \\) to \\( x e^x \\) gives \\( x \\).<\/p>\n\n\n\n
They are two sides of the same coin \u2014 one is \u201cfunction inverse in one direction,\u201d the other is \u201cfunction inverse in the other direction.\u201d<\/p>\n","protected":false},"excerpt":{"rendered":"
Confusing identities in lambert w function that W(x)e^W(x)=x=W(xe^x), how is this possible ? You\u2019ve hit on the most important \u2014 and initially confusing \u2014 property of the Lambert W function. Let\u2019s break it down step by step. — 1. Definition of \\( W \\) The Lambert \\( W \\) function is defined as the inverse […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_themeisle_gutenberg_block_has_review":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-177","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/posts\/177","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/comments?post=177"}],"version-history":[{"count":5,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/posts\/177\/revisions"}],"predecessor-version":[{"id":184,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/posts\/177\/revisions\/184"}],"wp:attachment":[{"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/media?parent=177"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/categories?post=177"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/potentsky.com\/math\/wp-json\/wp\/v2\/tags?post=177"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}