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Mirrors > Home > MPE Home > Th. List > df-atan | Structured version Visualization version GIF version |
Description: Define the arctangent function. See also remarks for df-asin 24592. Unlike arcsin and arccos, this function is not defined everywhere, because tan(𝑧) ≠ ±i for all 𝑧 ∈ ℂ. For all other 𝑧, there is a formula for arctan(𝑧) in terms of log, and we take that as the definition. Branch points are at ±i; branch cuts are on the pure imaginary axis not between -i and i, which is to say {𝑧 ∈ ℂ ∣ (i · 𝑧) ∈ (-∞, -1) ∪ (1, +∞)}. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
df-atan | ⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | catan 24591 | . 2 class arctan | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cc 9934 | . . . 4 class ℂ | |
4 | ci 9938 | . . . . . 6 class i | |
5 | 4 | cneg 10267 | . . . . 5 class -i |
6 | 5, 4 | cpr 4179 | . . . 4 class {-i, i} |
7 | 3, 6 | cdif 3571 | . . 3 class (ℂ ∖ {-i, i}) |
8 | c2 11070 | . . . . 5 class 2 | |
9 | cdiv 10684 | . . . . 5 class / | |
10 | 4, 8, 9 | co 6650 | . . . 4 class (i / 2) |
11 | c1 9937 | . . . . . . 7 class 1 | |
12 | 2 | cv 1482 | . . . . . . . 8 class 𝑥 |
13 | cmul 9941 | . . . . . . . 8 class · | |
14 | 4, 12, 13 | co 6650 | . . . . . . 7 class (i · 𝑥) |
15 | cmin 10266 | . . . . . . 7 class − | |
16 | 11, 14, 15 | co 6650 | . . . . . 6 class (1 − (i · 𝑥)) |
17 | clog 24301 | . . . . . 6 class log | |
18 | 16, 17 | cfv 5888 | . . . . 5 class (log‘(1 − (i · 𝑥))) |
19 | caddc 9939 | . . . . . . 7 class + | |
20 | 11, 14, 19 | co 6650 | . . . . . 6 class (1 + (i · 𝑥)) |
21 | 20, 17 | cfv 5888 | . . . . 5 class (log‘(1 + (i · 𝑥))) |
22 | 18, 21, 15 | co 6650 | . . . 4 class ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))) |
23 | 10, 22, 13 | co 6650 | . . 3 class ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))) |
24 | 2, 7, 23 | cmpt 4729 | . 2 class (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))) |
25 | 1, 24 | wceq 1483 | 1 wff arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))) |
Colors of variables: wff setvar class |
This definition is referenced by: atandm 24603 atanf 24607 atanval 24611 dvatan 24662 |
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