Definition df-log 26914

Description: Define the natural logarithm function on complex numbers. It is defined as the principal value, that is, the inverse of the exponential whose imaginary part lies in the interval (-pi, pi]. See http://en.wikipedia.org/wiki/Natural_logarithm and https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Paul Chapman, 21-Apr-2008.)

Assertion

Ref	Expression
df-log	⊢ log = ◡(exp ↾ (◡ℑ “ ( -π(,]π)))

Detailed syntax breakdown of Definition df-log

Step	Hyp	Ref	Expression
1		clog 26912	. 2 class log
2		ce 16256	. . . 4 class exp
3		cim 15294	. . . . . 6 class ℑ
4	3	ccnv 5654	. . . . 5 class ◡ℑ
5		cpi 16261	. . . . . . 7 class π
6	5	cneg 11569	. . . . . 6 class -π
7		cioc 13504	. . . . . 6 class (,]
8	6, 5, 7	co 7443	. . . . 5 class ( -π(,]π)
9	4, 8	cima 5658	. . . 4 class (◡ℑ “ ( -π(,]π))
10	2, 9	cres 5657	. . 3 class (exp ↾ (◡ℑ “ ( -π(,]π)))
11	10	ccnv 5654	. 2 class ◡(exp ↾ (◡ℑ “ ( -π(,]π)))
12	1, 11	wceq 1570	1 wff log = ◡(exp ↾ (◡ℑ “ ( -π(,]π)))

This definition is used by:  logrn  26916  dflog2  26918  dvlog  27009  efopnlem2  27015