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":{"ocean_post_layout":"","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"","ocean_second_sidebar":"","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"","ocean_custom_header_template":"","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"","ocean_menu_typo_font_family":"","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"default","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"default","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"","ocean_post_oembed":"","ocean_post_self_hosted_media":"","ocean_post_video_embed":"","ocean_link_format":"","ocean_link_format_target":"self","ocean_quote_format":"","ocean_quote_format_link":"post","ocean_gallery_link_images":"on","ocean_gallery_id":[],"footnotes":""},"categories":[1],"tags":[],"class_list":["post-177","post","type-post","status-publish","format-standard","hentry","category-uncategorized","entry"],"yoast_head":"\n