Definition df-asin

Description: Define the arcsine function. Because sin is not a one-to-one function, the literal inverse ` ``' sin is not a function. Rather than attempt to find the right domain on which to restrict sin in order to get a total function, we just define it in terms of log , which we already know is total (except at 0 ). There are branch points at -u 1 and 1 (at which the function is defined), and branch cuts along the real line not between -u 1 and 1 , which is to say ( -oo , -u 1 ) u. ( 1 , +oo ) ` . (Contributed by Mario Carneiro, 31-Mar-2015)

	Ref	Expression
Assertion	df-asin	⊢ arcsin = ( 𝑥 ∈ ℂ ↦ ( - i · ( log ‘ ( ( i · 𝑥 ) + ( √ ‘ ( 1 − ( 𝑥 ↑ 2 ) ) ) ) ) ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		casin	⊢ arcsin
1		vx	⊢ 𝑥
2		cc	⊢ ℂ
3		ci	⊢ i
4	3	cneg	⊢ - i
5		cmul	⊢ ·
6		clog	⊢ log
7	1	cv	⊢ 𝑥
8	3 7 5	co	⊢ ( i · 𝑥 )
9		caddc	⊢ +
10		csqrt	⊢ √
11		c1	⊢ 1
12		cmin	⊢ −
13		cexp	⊢ ↑
14		c2	⊢ 2
15	7 14 13	co	⊢ ( 𝑥 ↑ 2 )
16	11 15 12	co	⊢ ( 1 − ( 𝑥 ↑ 2 ) )
17	16 10	cfv	⊢ ( √ ‘ ( 1 − ( 𝑥 ↑ 2 ) ) )
18	8 17 9	co	⊢ ( ( i · 𝑥 ) + ( √ ‘ ( 1 − ( 𝑥 ↑ 2 ) ) ) )
19	18 6	cfv	⊢ ( log ‘ ( ( i · 𝑥 ) + ( √ ‘ ( 1 − ( 𝑥 ↑ 2 ) ) ) ) )
20	4 19 5	co	⊢ ( - i · ( log ‘ ( ( i · 𝑥 ) + ( √ ‘ ( 1 − ( 𝑥 ↑ 2 ) ) ) ) ) )
21	1 2 20	cmpt	⊢ ( 𝑥 ∈ ℂ ↦ ( - i · ( log ‘ ( ( i · 𝑥 ) + ( √ ‘ ( 1 − ( 𝑥 ↑ 2 ) ) ) ) ) ) )
22	0 21	wceq	⊢ arcsin = ( 𝑥 ∈ ℂ ↦ ( - i · ( log ‘ ( ( i · 𝑥 ) + ( √ ‘ ( 1 − ( 𝑥 ↑ 2 ) ) ) ) ) ) )