Definition df-atan 27083

Description: Define the arctangent function. See also remarks for df-asin 27081. 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.)

Assertion

Ref	Expression
df-atan	⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))

Detailed syntax breakdown of Definition df-atan

Step	Hyp	Ref	Expression
1		catan 27080	. 2 class arctan
2		vx	. . 3 setvar 𝑥
3		cc 11113	. . . 4 class ℂ
4		ci 11117	. . . . . 6 class i
5	4	cneg 11457	. . . . 5 class -i
6	5, 4	cpr 4593	. . . 4 class {-i, i}
7	3, 6	cdif 3903	. . 3 class (ℂ ∖ {-i, i})
8		c2 12310	. . . . 5 class 2
9		cdiv 11886	. . . . 5 class /
10	4, 8, 9	co 7419	. . . 4 class (i / 2)
11		c1 11116	. . . . . . 7 class 1
12	2	cv 1569	. . . . . . . 8 class 𝑥
13		cmul 11120	. . . . . . . 8 class ·
14	4, 12, 13	co 7419	. . . . . . 7 class (i · 𝑥)
15		cmin 11456	. . . . . . 7 class −
16	11, 14, 15	co 7419	. . . . . 6 class (1 − (i · 𝑥))
17		clog 26770	. . . . . 6 class log
18	16, 17	cfv 6540	. . . . 5 class (log‘(1 − (i · 𝑥)))
19		caddc 11118	. . . . . . 7 class +
20	11, 14, 19	co 7419	. . . . . 6 class (1 + (i · 𝑥))
21	20, 17	cfv 6540	. . . . 5 class (log‘(1 + (i · 𝑥)))
22	18, 21, 15	co 7419	. . . 4 class ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))
23	10, 22, 13	co 7419	. . 3 class ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))
24	2, 7, 23	cmpt 5194	. 2 class (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))
25	1, 24	wceq 1570	1 wff arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))

This definition is used by:  atandm  27092  atanf  27096  atanval  27100  dvatan  27151