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 e. CC |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		casin	|- arcsin
1		vx	|- x
2		cc	|- CC
3		ci	|- _i
4	3	cneg	|- -u _i
5		cmul	|- x.
6		clog	|- log
7	1	cv	|- x
8	3 7 5	co	|- ( _i x. x )
9		caddc	|- +
10		csqrt	|- sqrt
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	|- ( sqrt ` ( 1 - ( x ^ 2 ) ) )
18	8 17 9	co	|- ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) )
19	18 6	cfv	|- ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) )
20	4 19 5	co	|- ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) )
21	1 2 20	cmpt	|- ( x e. CC |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) )
22	0 21	wceq	|- arcsin = ( x e. CC |-> ( -u _i x. ( log ` ( ( _i x. x ) + ( sqrt ` ( 1 - ( x ^ 2 ) ) ) ) ) ) )