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 ArcSin

 http://functions.wolfram.com/01.12.16.0208.01

 Input Form

 a ArcSin[x] + b ArcSinh[y] == (-1)^(Floor[1/2 - Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a)]/ Pi] - Floor[-(Arg[((-I) (y + Sqrt[y^2 + 1])^b)/(I x + Sqrt[1 - x^2])^ (I a) - 1]/(2 Pi)) - Arg[(I (y + Sqrt[y^2 + 1])^b)/(I x + Sqrt[1 - x^2])^(I a) - 1]/ (2 Pi) + 1/2]) ArcSinh[((1/2) (I x + Sqrt[1 - x^2])^(I a) ((y + Sqrt[y^2 + 1])^(2 b)/(I x + Sqrt[1 - x^2])^(2 I a) - 1))/ (y + Sqrt[y^2 + 1])^b] - (1/2) I (-1)^(-Floor[-(Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a)]/ Pi)] - Floor[ -(Arg[((-I) (y + Sqrt[y^2 + 1])^b)/(I x + Sqrt[1 - x^2])^(I a) - 1]/ (2 Pi)) - Arg[(I (y + Sqrt[y^2 + 1])^b)/(I x + Sqrt[1 - x^2])^ (I a) - 1]/(2 Pi) + 1/2]) (1 - (-1)^Floor[1/2 - Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^ (I a)]/Pi] + (-1)^(Floor[-(Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a)]/ Pi)] + Floor[ -(Arg[((-I) (y + Sqrt[y^2 + 1])^b)/(I x + Sqrt[1 - x^2])^(I a) - 1]/ (2 Pi)) - Arg[(I (y + Sqrt[y^2 + 1])^b)/(I x + Sqrt[1 - x^2])^ (I a) - 1]/(2 Pi) + 1/2] + Floor[Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a)]/(2 Pi) - Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a) - I]/(2 Pi) + 1/2] + Floor[-(Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a)]/ (2 Pi)) + Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a) - I]/(2 Pi) + 1/2]) - (-1)^(Floor[-(Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a)]/ Pi)] + Floor[ -(Arg[((-I) (y + Sqrt[y^2 + 1])^b)/(I x + Sqrt[1 - x^2])^(I a) - 1]/ (2 Pi)) - Arg[(I (y + Sqrt[y^2 + 1])^b)/(I x + Sqrt[1 - x^2])^ (I a) - 1]/(2 Pi) + 1/2] + Floor[Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a)]/(2 Pi) - Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a) + I]/(2 Pi) + 1/2] + Floor[-(Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a)]/ (2 Pi)) + Arg[(y + Sqrt[y^2 + 1])^b/(I x + Sqrt[1 - x^2])^(I a) + I]/(2 Pi) + 1/2])) Pi - 2 I Pi (Floor[(-Arg[(I x + Sqrt[1 - x^2])^((-I) a)] - Arg[(y + Sqrt[y^2 + 1])^b] + Pi)/(2 Pi)] + Floor[(Pi - Im[b Log[y + Sqrt[y^2 + 1]]])/(2 Pi)] + Floor[(Re[a Log[I x + Sqrt[1 - x^2]]] + Pi)/(2 Pi)])

 Standard Form

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 MathML Form

 a sin - 1 ( x ) + b sinh - 1 ( y ) ( - 1 ) 1 2 - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b ) π - - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b - 1 ) 2 π + 1 2 - arg ( ( y + y 2 + 1 ) b ( - ) ( x + 1 - x 2 ) - a - 1 ) 2 π sinh - 1 ( 1 2 ( x + 1 - x 2 ) a ( y + y 2 + 1 ) - b ( ( x + 1 - x 2 ) - 2 a ( y + y 2 + 1 ) 2 b - 1 ) ) - 1 2 ( - 1 ) - - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b - 1 ) 2 π + 1 2 - arg ( ( y + y 2 + 1 ) b ( - ) ( x + 1 - x 2 ) - a - 1 ) 2 π - - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b ) π ( 1 - ( - 1 ) 1 2 - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b ) π + ( - 1 ) - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b ) 2 π + arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b - ) 2 π + 1 2 + - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b - 1 ) 2 π + 1 2 - arg ( ( y + y 2 + 1 ) b ( - ) ( x + 1 - x 2 ) - a - 1 ) 2 π + arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b ) 2 π + 1 2 - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b - ) 2 π + - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b ) π - ( - 1 ) - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b ) 2 π + arg ( ( y + y 2 + 1 ) b ( x + 1 - x 2 ) - a + ) 2 π + 1 2 + - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b - 1 ) 2 π + 1 2 - arg ( ( y + y 2 + 1 ) b ( - ) ( x + 1 - x 2 ) - a - 1 ) 2 π + arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b ) 2 π + 1 2 - arg ( ( y + y 2 + 1 ) b ( x + 1 - x 2 ) - a + ) 2 π + - arg ( ( x + 1 - x 2 ) - a ( y + y 2 + 1 ) b ) π ) π - 2 π ( - arg ( ( y + y 2 + 1 ) b ) - arg ( ( x + 1 - x 2 ) - a ) + π 2 π + Re ( a log ( x + 1 - x 2 ) ) + π 2 π + π - Im ( b log ( y + y 2 + 1 ) ) 2 π ) a x b y -1 1 2 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 -1 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 2 -1 1 2 -1 y y 2 1 1 2 b -1 x 1 -1 x 2 1 2 -1 a -1 2 -1 1 2 x 1 -1 x 2 1 2 a y y 2 1 1 2 -1 b x 1 -1 x 2 1 2 -2 a y y 2 1 1 2 2 b -1 -1 1 2 -1 -1 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 2 -1 1 2 -1 y y 2 1 1 2 b -1 x 1 -1 x 2 1 2 -1 a -1 2 -1 -1 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 1 -1 -1 1 2 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 -1 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b 2 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 2 -1 1 2 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 2 -1 1 2 -1 y y 2 1 1 2 b -1 x 1 -1 x 2 1 2 -1 a -1 2 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b 2 -1 1 2 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 2 -1 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 -1 -1 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b 2 -1 y y 2 1 1 2 b x 1 -1 x 2 1 2 -1 a 2 -1 1 2 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 2 -1 1 2 -1 y y 2 1 1 2 b -1 x 1 -1 x 2 1 2 -1 a -1 2 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b 2 -1 1 2 -1 y y 2 1 1 2 b x 1 -1 x 2 1 2 -1 a 2 -1 -1 x 1 -1 x 2 1 2 -1 a y y 2 1 1 2 b -1 -1 2 -1 y y 2 1 1 2 b -1 x 1 -1 x 2 1 2 -1 a 2 -1 a x 1 -1 x 2 1 2 2 -1 -1 b y y 2 1 1 2 2 -1 [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2007-05-02