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	$$⊢arcsin=\left(x\in ℂ⟼\left(-\mathrm{i}\right)\log \left(\mathrm{i}x+\sqrt{1-x^2}\right)\right)$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		casin	$$classarcsin$$
1		vx	$$setvarx$$
2		cc	$$classℂ$$
3		ci	$$class\mathrm{i}$$
4	3	cneg	$$class\left(-\mathrm{i}\right)$$
5		cmul	$$class\times$$
6		clog	$$classlog$$
7	1	cv	$$setvarx$$
8	3 7 5	co	$$class\mathrm{i}x$$
9		caddc	$$class+$$
10		csqrt	$$class\surd$$
11		c1	$$class1$$
12		cmin	$$class-$$
13		cexp	$$class^$$
14		c2	$$class2$$
15	7 14 13	co	$$class{x}^2$$
16	11 15 12	co	$$class\left(1-x^2\right)$$
17	16 10	cfv	$$class\sqrt{1-x^2}$$
18	8 17 9	co	$$class\left(\mathrm{i}x+\sqrt{1-x^2}\right)$$
19	18 6	cfv	$$class\log \left(\mathrm{i}x+\sqrt{1-x^2}\right)$$
20	4 19 5	co	$$class\left(-\mathrm{i}\right)\log \left(\mathrm{i}x+\sqrt{1-x^2}\right)$$
21	1 2 20	cmpt	$$class\left(x\in ℂ⟼\left(-\mathrm{i}\right)\log \left(\mathrm{i}x+\sqrt{1-x^2}\right)\right)$$
22	0 21	wceq	$$wffarcsin=\left(x\in ℂ⟼\left(-\mathrm{i}\right)\log \left(\mathrm{i}x+\sqrt{1-x^2}\right)\right)$$