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Definition df-log 23506
Description: Define the natural logarithm function on complex numbers. See http://en.wikipedia.org/wiki/Natural_logarithm ("The natural logarithm function can also be defined as the inverse function of the exponential function"). To obtain a function, only the principle value of the multivalued inverses of the exponential function, i.e. the inverse whose imaginary part lies in the interval (-pi, pi], see https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Paul Chapman, 21-Apr-2008.)
Assertion
Ref Expression
df-log  |-  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )

Detailed syntax breakdown of Definition df-log
StepHypRef Expression
1 clog 23504 . 2  class  log
2 ce 14114 . . . 4  class  exp
3 cim 13161 . . . . . 6  class  Im
43ccnv 4833 . . . . 5  class  `' Im
5 cpi 14119 . . . . . . 7  class  pi
65cneg 9861 . . . . . 6  class  -u pi
7 cioc 11636 . . . . . 6  class  (,]
86, 5, 7co 6290 . . . . 5  class  ( -u pi (,] pi )
94, 8cima 4837 . . . 4  class  ( `' Im " ( -u pi (,] pi ) )
102, 9cres 4836 . . 3  class  ( exp  |`  ( `' Im "
( -u pi (,] pi ) ) )
1110ccnv 4833 . 2  class  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
121, 11wceq 1444 1  wff  log  =  `' ( exp  |`  ( `' Im " ( -u pi (,] pi ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  logrn  23508  dflog2  23510  dvlog  23596  efopnlem2  23602
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