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http://functions.wolfram.com/01.12.16.0206.01
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a ArcSin[x] + b ArcSec[y] ==
(Pi b)/2 - 2 I Pi (Floor[(-Arg[(I x + Sqrt[1 - x^2])^((-I) a)] -
Arg[(Sqrt[1 - 1/y^2] + I/y)^(I b)] + Pi)/(2 Pi)] +
Floor[(Re[a Log[I x + Sqrt[1 - x^2]]] + Pi)/(2 Pi)] +
Floor[(Pi - Re[b Log[Sqrt[1 - 1/y^2] + I/y]])/(2 Pi)]) +
I Pi
(1 - (-1)^(Floor[-(Arg[(Sqrt[1 - 1/y^2] + I/y)^(I b)/(I x + Sqrt[1 - x^2])^
(I a) + 1]/(2 Pi))] -
Floor[-(Arg[(Sqrt[1 - 1/y^2] + I/y)^(I b)/(I x + Sqrt[1 - x^2])^(I a)]/
(2 Pi))])) -
I (-1)^(Floor[Arg[(Sqrt[1 - 1/y^2] + I/y)^(I b)/(I x + Sqrt[1 - x^2])^
(I a) - 1]/(2 Pi) -
Arg[(Sqrt[1 - 1/y^2] + I/y)^(I b)/(I x + Sqrt[1 - x^2])^(I a) + 1]/
(2 Pi) + 1/2] +
Floor[-(Arg[(Sqrt[1 - 1/y^2] + I/y)^(I b)/(I x + Sqrt[1 - x^2])^(I a)]/
Pi)]) ArcSec[(2 (Sqrt[1 - 1/y^2] + I/y)^(I b))/
((I x + Sqrt[1 - x^2])^(I a) ((Sqrt[1 - 1/y^2] + I/y)^(2 I b)/
(I x + Sqrt[1 - x^2])^(2 I a) + 1))]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sec </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> 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</msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> </mrow> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> π </mi> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> 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Date Added to functions.wolfram.com (modification date)
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