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Mirrors > Home > MPE Home > Th. List > df-log | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-log | ⊢ log = ◡(exp ↾ (◡ℑ “ (-π(,]π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clog 24105 | . 2 class log | |
2 | ce 14631 | . . . 4 class exp | |
3 | cim 13686 | . . . . . 6 class ℑ | |
4 | 3 | ccnv 5037 | . . . . 5 class ◡ℑ |
5 | cpi 14636 | . . . . . . 7 class π | |
6 | 5 | cneg 10146 | . . . . . 6 class -π |
7 | cioc 12047 | . . . . . 6 class (,] | |
8 | 6, 5, 7 | co 6549 | . . . . 5 class (-π(,]π) |
9 | 4, 8 | cima 5041 | . . . 4 class (◡ℑ “ (-π(,]π)) |
10 | 2, 9 | cres 5040 | . . 3 class (exp ↾ (◡ℑ “ (-π(,]π))) |
11 | 10 | ccnv 5037 | . 2 class ◡(exp ↾ (◡ℑ “ (-π(,]π))) |
12 | 1, 11 | wceq 1475 | 1 wff log = ◡(exp ↾ (◡ℑ “ (-π(,]π))) |
Colors of variables: wff setvar class |
This definition is referenced by: logrn 24109 dflog2 24111 dvlog 24197 efopnlem2 24203 |
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