Post Reply 
0^0 = 1?
04-23-2017, 02:58 PM
Post: #13
RE: [wp34s] 0^0 = 1?
Well, yes, y^x is a bivariate function, so you can come at (0, 0) from more than just two directions. If we parameterize the direction as an angle φ from the positive x axis, we get

y^x = (r sin(φ))^(r cos(φ))

and the value we're looking for is the limit of that as r goes to zero.
For all φ ∈ (0, π), so positive y, this limit is 1; for all φ ∈ (π, 2π), so negative y, this limit is also 1, albeit with a side trip through the complex plane.

This leaves only the cases φ = 0 and φ = π, i.e. y = 0 with x > 0 and x < 0, respectively. Lim (x->0) 0^x is undefined if you come at it from below, because of division by 0. From above it's tricky, because while 0^x = 0 for all x > 0, suggesting simply continuing this line yielding 0^0 = 0, its behavior at x = 0 is rather special:

0^x = e^(x * log(0)) = e^(x * -Inf)

because x * -Inf = -Inf for all x > 0, and undefined when x = 0. The usual tricks for resolving limits by dropping higher-order terms or cancelling factors don't work here because that infinity really is a hard infinity, not a factor that goes to inifinity as we approach the limit.

So, lim (r->0) (r sin(φ))^(r cos(φ)) = 1 for φ ∈ (0, π) ∪ (π, 2π) and undefined for φ = 0 and φ = π.

Make of that what you will. :-)
Visit this user's website Find all posts by this user
Quote this message in a reply
Post Reply 


Messages In This Thread
0^0 = 1? - emece67 - 04-21-2017, 07:43 AM
RE: [wp34s] 0^0 = 1? - Werner - 04-21-2017, 09:55 AM
RE: [wp34s] 0^0 = 1? - Thomas Okken - 04-21-2017, 01:54 PM
RE: [wp34s] 0^0 = 1? - Egan Ford - 04-22-2017, 01:02 AM
RE: [wp34s] 0^0 = 1? - Chris Dreher - 04-22-2017, 05:52 PM
RE: [wp34s] 0^0 = 1? - Werner - 04-22-2017, 08:00 AM
RE: [wp34s] 0^0 = 1? - Thomas Okken - 04-22-2017, 11:35 AM
RE: [wp34s] 0^0 = 1? - emece67 - 04-22-2017, 09:43 AM
RE: [wp34s] 0^0 = 1? - AlexFekken - 04-22-2017, 10:40 PM
RE: [wp34s] 0^0 = 1? - Thomas Okken - 04-22-2017, 11:28 PM
RE: [wp34s] 0^0 = 1? - AlexFekken - 04-23-2017, 01:06 AM
RE: [wp34s] 0^0 = 1? - AlexFekken - 04-23-2017, 01:46 AM
RE: [wp34s] 0^0 = 1? - Thomas Okken - 04-23-2017 02:58 PM
RE: [wp34s] 0^0 = 1? - Claudio L. - 04-23-2017, 07:33 PM
RE: [wp34s] 0^0 = 1? - Thomas Okken - 04-23-2017, 07:45 PM
RE: [wp34s] 0^0 = 1? - Claudio L. - 04-24-2017, 12:40 AM
RE: [wp34s] 0^0 = 1? - Thomas Okken - 04-24-2017, 02:09 AM
RE: [wp34s] 0^0 = 1? - BobVA - 04-23-2017, 04:41 PM
RE: [wp34s] 0^0 = 1? - Chris Dreher - 04-24-2017, 03:45 AM
RE: [wp34s] 0^0 = 1? - Thomas Okken - 04-24-2017, 04:26 AM
RE: [wp34s] 0^0 = 1? - Chris Dreher - 04-24-2017, 06:24 AM
RE: [wp34s] 0^0 = 1? - AlexFekken - 04-25-2017, 11:14 AM
RE: [wp34s] 0^0 = 1? - Felix Gross - 04-25-2017, 03:00 PM
RE: [wp34s] 0^0 = 1? - emece67 - 04-25-2017, 07:53 PM
RE: [wp34s] 0^0 = 1? - EdS2 - 04-26-2017, 06:03 AM
RE: [wp34s] 0^0 = 1? - Chris Dreher - 04-27-2017, 04:55 AM
RE: [wp34s] 0^0 = 1? - Paul Dale - 04-26-2017, 06:40 AM
RE: 0^0 = 1? - emece67 - 05-03-2017, 03:24 PM
RE: [wp34s] 0^0 = 1? - Gerald H - 04-27-2017, 05:48 AM
RE: [wp34s] 0^0 = 1? - d b - 04-27-2017, 03:35 PM
RE: [wp34s] 0^0 = 1? - rprosperi - 04-27-2017, 04:28 PM
RE: [wp34s] 0^0 = 1? - emece67 - 04-27-2017, 05:32 PM
RE: 0^0 = 1? - Felix Gross - 04-27-2017, 08:01 PM
RE: 0^0 = 1? - Dwight Sturrock - 04-27-2017, 11:20 PM
RE: 0^0 = 1? - Craig Bladow - 04-27-2017, 11:46 PM
RE: 0^0 = 1? - matthiaspaul - 04-27-2017, 11:51 PM
RE: 0^0 = 1? - ttw - 04-28-2017, 04:57 AM
RE: 0^0 = 1? - Chris Dreher - 04-28-2017, 04:50 PM
RE: 0^0 = 1? - Han - 04-28-2017, 05:50 PM
RE: 0^0 = 1? - matthiaspaul - 04-28-2017, 10:22 PM
RE: 0^0 = 1? - pier4r - 04-28-2017, 07:39 AM
RE: 0^0 = 1? - AndiGer - 04-28-2017, 07:18 PM
RE: 0^0 = 1? - ttw - 04-30-2017, 12:35 AM
RE: 0^0 = 1? - brickviking - 04-30-2017, 04:59 AM
RE: 0^0 = 1? - Anderson Costa - 05-09-2017, 01:04 AM
RE: 0^0 = 1? - Felix Gross - 05-09-2017, 07:30 PM



User(s) browsing this thread: 1 Guest(s)