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Iteration basics
#2
Common difficulty. Sometime brain blocks...

Assume
Code:
´
f(x) = x^2+2
Code:
´
a) f(x)^2 = (x^2+2)^2           // taking f to the 2'nd power
          =  x^4 + 4x^2 +4

b) f°2(x) = f(f(x))             // iteration of f
          = (f(x))^2  + 2
          =  (x^4 + 4x^2 +4) + 2
          = x^4+4x^2+6

c)  e^f(x) = ...                // exponentiation of f

Assume
Code:
´
       f(x) = e^x
            = 1 + x/1! + x^2/2! + ....
Code:
´
a)       f(x)^2 = (1 + x/1! + x^2/2! +...)^2   // taking f to the second power
                =  1 + 2*x/1! + 4*x^2/2! + 8*x^3/3! + 16*x^4/4! +...
                = f(2x)

b)       f°2(x) = f(f(x))                      // iteration of f
                = f(e^x)
                = 1 + e^x/1! + e^2x/2! + ...

c)       e^f(x) = ...                          // exponentiation of f
                     // is in this case equal to second iteration

(You had this right previous times Smile )

Gottfried
Gottfried Helms, Kassel
Reply


Messages In This Thread
Iteration basics - by Ivars - 03/20/2008, 10:34 AM
RE: Iteration basics - by Ivars - 03/20/2008, 05:24 PM
RE: Iteration basics - by Gottfried - 03/21/2008, 08:37 AM
RE: Iteration basics - by Ivars - 04/03/2008, 08:05 AM
RE: Iteration basics - by Gottfried - 04/03/2008, 08:19 AM
RE: Iteration basics - by Ivars - 04/03/2008, 10:13 AM
RE: Iteration basics - by andydude - 04/05/2008, 11:35 PM
RE: Iteration basics - by Gottfried - 04/06/2008, 07:16 AM
RE: Iteration basics - by Ivars - 04/06/2008, 08:55 AM
RE: Iteration basics - by andydude - 04/07/2008, 12:01 AM
RE: Iteration basics - by Ivars - 04/20/2008, 10:15 PM
RE: Iteration basics - by bo198214 - 04/21/2008, 08:21 PM
RE: Iteration basics - by andydude - 04/22/2008, 05:30 AM
RE: Iteration basics - by Ivars - 04/22/2008, 07:02 AM
RE: Iteration basics - by Ivars - 04/09/2008, 07:56 PM
RE: Iteration basics - by Ivars - 05/09/2008, 09:45 AM
RE: Iteration basics - by bo198214 - 05/09/2008, 02:49 PM
RE: Iteration basics - by Ivars - 05/27/2008, 10:33 AM
RE: Iteration basics - by Gottfried - 05/27/2008, 07:35 PM
RE: Iteration basics - by Ivars - 05/30/2008, 06:08 AM
RE: Iteration basics - by Xorter - 01/02/2017, 05:21 PM
RE: Imaginary iterates of exponentiation - by Gottfried - 03/20/2008, 12:04 PM

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