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Definition df-atan 24489
Description: Define the arctangent function. See also remarks for df-asin 24487. 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
StepHypRef Expression
1 catan 24486 . 2 class arctan
2 vx . . 3 setvar 𝑥
3 cc 9879 . . . 4 class
4 ci 9883 . . . . . 6 class i
54cneg 10212 . . . . 5 class -i
65, 4cpr 4155 . . . 4 class {-i, i}
73, 6cdif 3557 . . 3 class (ℂ ∖ {-i, i})
8 c2 11015 . . . . 5 class 2
9 cdiv 10629 . . . . 5 class /
104, 8, 9co 6605 . . . 4 class (i / 2)
11 c1 9882 . . . . . . 7 class 1
122cv 1479 . . . . . . . 8 class 𝑥
13 cmul 9886 . . . . . . . 8 class ·
144, 12, 13co 6605 . . . . . . 7 class (i · 𝑥)
15 cmin 10211 . . . . . . 7 class
1611, 14, 15co 6605 . . . . . 6 class (1 − (i · 𝑥))
17 clog 24200 . . . . . 6 class log
1816, 17cfv 5850 . . . . 5 class (log‘(1 − (i · 𝑥)))
19 caddc 9884 . . . . . . 7 class +
2011, 14, 19co 6605 . . . . . 6 class (1 + (i · 𝑥))
2120, 17cfv 5850 . . . . 5 class (log‘(1 + (i · 𝑥)))
2218, 21, 15co 6605 . . . 4 class ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))
2310, 22, 13co 6605 . . 3 class ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))
242, 7, 23cmpt 4678 . 2 class (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))
251, 24wceq 1480 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  24498  atanf  24502  atanval  24506  dvatan  24557
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