Expression 88: "C" Subscript, "o" "t" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, negative StartFraction, sin left parenthesis, 2 "z" . "x" , right parenthesis Over cos left parenthesis, 2 "z" . "x" , right parenthesis minus cosh left parenthesis, 2 "z" . "y" , right parenthesis , EndFraction , StartFraction, sinh left parenthesis, 2 "z" . "y" , right parenthesis Over cos left parenthesis, 2 "z" . "x" , right parenthesis minus cosh left parenthesis, 2 "z" . "y" , right parenthesis , EndFraction , right parenthesisCotz=−sin2z.xcos2z.x−cosh2z.y,sinh2z.ycos2z.x−cosh2z.y
88
𝘐𝘯𝘷𝘦𝘳𝘴𝘦 𝘛𝘳𝘪𝘨𝘰𝘯𝘰𝘮𝘦𝘵𝘳𝘪𝘤 𝘍𝘶𝘯𝘤𝘵𝘪𝘰𝘯𝘴
89
Expression 90: "A" Subscript, "r" "c" "S" "i" "n" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, arctan left parenthesis, Root Index 4, Start Root, 4 "z" . "x" squared "z" . "y" squared plus left parenthesis, 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , End Root sin left parenthesis, 0.5arctan left parenthesis, negative 2 "z" . "x" "z" . "y" , 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis , right parenthesis plus "z" . "x" , Root Index 4, Start Root, 4 "z" . "x" squared "z" . "y" squared plus left parenthesis, 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , End Root cos left parenthesis, 0.5arctan left parenthesis, negative 2 "z" . "x" "z" . "y" , 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis , right parenthesis minus "z" . "y" , right parenthesis , negative 0.5ln left parenthesis, left parenthesis, Root Index 4, Start Root, 4 "z" . "x" squared "z" . "y" squared plus left parenthesis, 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , End Root cos left parenthesis, 0.5arctan left parenthesis, negative 2 "z" . "x" "z" . "y" , 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis , right parenthesis minus "z" . "y" , right parenthesis squared positive left parenthesis, Root Index 4, Start Root, 4 "z" . "x" squared "z" . "y" squared plus left parenthesis, 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , End Root sin left parenthesis, 0.5arctan left parenthesis, negative 2 "z" . "x" "z" . "y" , 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis , right parenthesis plus "z" . "x" , right parenthesis squared , right parenthesis , right parenthesisArcSinz=arctan44z.x2z.y2+1−z.x2+z.y22sin0.5arctan−2z.xz.y,1−z.x2+z.y2+z.x,44z.x2z.y2+1−z.x2+z.y22cos0.5arctan−2z.xz.y,1−z.x2+z.y2−z.y,−0.5ln44z.x2z.y2+1−z.x2+z.y22cos0.5arctan−2z.xz.y,1−z.x2+z.y2−z.y2+44z.x2z.y2+1−z.x2+z.y22sin0.5arctan−2z.xz.y,1−z.x2+z.y2+z.x2
90
Expression 91: "A" Subscript, "r" "c" "C" "o" "s" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, arcsin left parenthesis, 1 , right parenthesis minus arctan left parenthesis, Root Index 4, Start Root, 4 "z" . "x" squared "z" . "y" squared plus left parenthesis, 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , End Root sin left parenthesis, 0.5arctan left parenthesis, negative 2 "z" . "x" "z" . "y" , 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis , right parenthesis plus "z" . "x" , Root Index 4, Start Root, 4 "z" . "x" squared "z" . "y" squared plus left parenthesis, 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , End Root cos left parenthesis, 0.5arctan left parenthesis, negative 2 "z" . "x" "z" . "y" , 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis , right parenthesis minus "z" . "y" , right parenthesis , 0.5ln left parenthesis, left parenthesis, "z" . "y" minus Root Index 4, Start Root, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared minus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared plus 1 , right parenthesis squared , End Root cos left parenthesis, 0.5arctan left parenthesis, negative 2 "z" . "x" "z" . "y" , 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis , right parenthesis , right parenthesis squared positive left parenthesis, "z" . "x" plus Root Index 4, Start Root, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared minus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared plus 1 , right parenthesis squared , End Root sin left parenthesis, 0.5arctan left parenthesis, negative 2 "z" . "x" "z" . "y" , 1 minus "z" . "x" squared plus "z" . "y" squared , right parenthesis , right parenthesis , right parenthesis squared , right parenthesis , right parenthesisArcCosz=arcsin1−arctan44z.x2z.y2+1−z.x2+z.y22sin0.5arctan−2z.xz.y,1−z.x2+z.y2+z.x,44z.x2z.y2+1−z.x2+z.y22cos0.5arctan−2z.xz.y,1−z.x2+z.y2−z.y,0.5lnz.y−4z.x4+2z.y2−1z.x2+z.y2+12cos0.5arctan−2z.xz.y,1−z.x2+z.y22+z.x+4z.x4+2z.y2−1z.x2+z.y2+12sin0.5arctan−2z.xz.y,1−z.x2+z.y22
91
Expression 92: "A" Subscript, "r" "c" "T" "a" "n" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, 0.5 left parenthesis, arctan left parenthesis, "z" . "x" , 1 minus "z" . "y" , right parenthesis minus arctan left parenthesis, negative "z" . "x" , "z" . "y" plus 1 , right parenthesis , right parenthesis , 0.2 5ln left parenthesis, StartFraction, left parenthesis, "z" . "y" plus 1 , right parenthesis squared plus "z" . "x" squared Over "z" . "x" squared plus left parenthesis, 1 minus "z" . "y" , right parenthesis squared , EndFraction , right parenthesis , right parenthesisArcTanz=0.5arctanz.x,1−z.y−arctan−z.x,z.y+1,0.25lnz.y+12+z.x2z.x2+1−z.y2
92
Expression 93: "A" Subscript, "r" "c" "C" "s" "c" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, arctan left parenthesis, Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared minus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared plus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root sin left parenthesis, 0.5arctan left parenthesis, StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "y" squared minus "z" . "x" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis plus StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared minus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared plus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root cos left parenthesis, 0.5arctan left parenthesis, StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "y" squared minus "z" . "x" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis plus StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesis , negative 0.5ln left parenthesis, left parenthesis, Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared minus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared plus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root cos left parenthesis, 0.5arctan left parenthesis, StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "y" squared minus "z" . "x" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis plus StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesis squared plus left parenthesis, Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared minus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared plus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root sin left parenthesis, 0.5arctan left parenthesis, StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "y" squared minus "z" . "x" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis plus StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesis squared , right parenthesis , right parenthesisArcCscz=arctan4z.x4+2z.y2−1z.x2+z.y2+12z.x2+z.y22sin0.5arctan2z.xz.yz.x2+z.y22,z.y2−z.x2z.x2+z.y22+1+z.xz.x2+z.y2,4z.x4+2z.y2−1z.x2+z.y2+12z.x2+z.y22cos0.5arctan2z.xz.yz.x2+z.y22,z.y2−z.x2z.x2+z.y22+1+z.yz.x2+z.y2,−0.5ln4z.x4+2z.y2−1z.x2+z.y2+12z.x2+z.y22cos0.5arctan2z.xz.yz.x2+z.y22,z.y2−z.x2z.x2+z.y22+1+z.yz.x2+z.y22+4z.x4+2z.y2−1z.x2+z.y2+12z.x2+z.y22sin0.5arctan2z.xz.yz.x2+z.y22,z.y2−z.x2z.x2+z.y22+1+z.xz.x2+z.y22
93
Expression 94: "A" Subscript, "r" "c" "S" "e" "c" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, arcsin left parenthesis, 1 , right parenthesis minus arctan left parenthesis, Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared minus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared plus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root sin left parenthesis, 0.5arctan left parenthesis, StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "y" squared minus "z" . "x" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis plus StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared minus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared plus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root cos left parenthesis, 0.5arctan left parenthesis, StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "y" squared minus "z" . "x" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis plus StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesis , 0.5ln left parenthesis, left parenthesis, Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared minus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared plus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root cos left parenthesis, 0.5arctan left parenthesis, StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "y" squared minus "z" . "x" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis plus StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesis squared plus left parenthesis, Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared minus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared plus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root sin left parenthesis, 0.5arctan left parenthesis, StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "y" squared minus "z" . "x" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis plus StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesis squared , right parenthesis , right parenthesisArcSecz=arcsin1−arctan4z.x4+2z.y2−1z.x2+z.y2+12z.x2+z.y22sin0.5arctan2z.xz.yz.x2+z.y22,z.y2−z.x2z.x2+z.y22+1+z.xz.x2+z.y2,4z.x4+2z.y2−1z.x2+z.y2+12z.x2+z.y22cos0.5arctan2z.xz.yz.x2+z.y22,z.y2−z.x2z.x2+z.y22+1+z.yz.x2+z.y2,0.5ln4z.x4+2z.y2−1z.x2+z.y2+12z.x2+z.y22cos0.5arctan2z.xz.yz.x2+z.y22,z.y2−z.x2z.x2+z.y22+1+z.yz.x2+z.y22+4z.x4+2z.y2−1z.x2+z.y2+12z.x2+z.y22sin0.5arctan2z.xz.yz.x2+z.y22,z.y2−z.x2z.x2+z.y22+1+z.xz.x2+z.y22
94
Expression 95: "A" Subscript, "r" "c" "C" "o" "t" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, 0.5 left parenthesis, arctan left parenthesis, StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction plus 1 , right parenthesis minus arctan left parenthesis, negative StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , 1 minus StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesis , right parenthesis , 0.2 5ln left parenthesis, StartFraction, "z" . "x" squared plus left parenthesis, "z" . "y" plus 1 , right parenthesis squared Over "z" . "x" squared plus left parenthesis, "z" . "y" minus 1 , right parenthesis squared , EndFraction , right parenthesis , right parenthesisArcCotz=0.5arctanz.xz.x2+z.y2,z.yz.x2+z.y2+1−arctan−z.xz.x2+z.y2,1−z.yz.x2+z.y2,0.25lnz.x2+z.y+12z.x2+z.y−12
95
𝘏𝘺𝘱𝘦𝘳𝘣𝘰𝘭𝘪𝘤 𝘍𝘶𝘯𝘤𝘵𝘪𝘰𝘯𝘴
96
Expression 97: "S" Subscript, "i" "n" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, cos left parenthesis, "z" . "y" , right parenthesis sinh left parenthesis, "z" . "x" , right parenthesis , sin left parenthesis, "z" . "y" , right parenthesis cosh left parenthesis, "z" . "x" , right parenthesis , right parenthesisSinhz=cosz.ysinhz.x,sinz.ycoshz.x
97
Expression 98: "C" Subscript, "o" "s" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, cos left parenthesis, "z" . "y" , right parenthesis cosh left parenthesis, "z" . "x" , right parenthesis , sin left parenthesis, "z" . "y" , right parenthesis sinh left parenthesis, "z" . "x" , right parenthesis , right parenthesisCoshz=cosz.ycoshz.x,sinz.ysinhz.x
98
Expression 99: "T" Subscript, "a" "n" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, StartFraction, sinh left parenthesis, 2 "z" . "x" , right parenthesis Over cos left parenthesis, 2 "z" . "y" , right parenthesis plus cosh left parenthesis, 2 "z" . "x" , right parenthesis , EndFraction , StartFraction, sin left parenthesis, 2 "z" . "y" , right parenthesis Over cos left parenthesis, 2 "z" . "y" , right parenthesis plus cosh left parenthesis, 2 "z" . "x" , right parenthesis , EndFraction , right parenthesisTanhz=sinh2z.xcos2z.y+cosh2z.x,sin2z.ycos2z.y+cosh2z.x
99
Expression 100: "C" Subscript, "s" "c" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, negative StartFraction, 2cos left parenthesis, "z" . "y" , right parenthesis sinh left parenthesis, "z" . "x" , right parenthesis Over cos left parenthesis, 2 "z" . "y" , right parenthesis minus cosh left parenthesis, 2 "z" . "x" , right parenthesis , EndFraction , StartFraction, 2sin left parenthesis, "z" . "y" , right parenthesis cosh left parenthesis, "z" . "x" , right parenthesis Over cos left parenthesis, 2 "z" . "y" , right parenthesis minus cosh left parenthesis, 2 "z" . "x" , right parenthesis , EndFraction , right parenthesisCschz=−2cosz.ysinhz.xcos2z.y−cosh2z.x,2sinz.ycoshz.xcos2z.y−cosh2z.x
100
Expression 101: "S" Subscript, "e" "c" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, StartFraction, 2cos left parenthesis, "z" . "y" , right parenthesis cosh left parenthesis, "z" . "x" , right parenthesis Over cos left parenthesis, 2 "z" . "y" , right parenthesis plus cosh left parenthesis, 2 "z" . "x" , right parenthesis , EndFraction , negative StartFraction, 2sin left parenthesis, "z" . "y" , right parenthesis sinh left parenthesis, "z" . "x" , right parenthesis Over cos left parenthesis, 2 "z" . "y" , right parenthesis plus cosh left parenthesis, 2 "z" . "x" , right parenthesis , EndFraction , right parenthesisSechz=2cosz.ycoshz.xcos2z.y+cosh2z.x,−2sinz.ysinhz.xcos2z.y+cosh2z.x
101
Expression 102: "C" Subscript, "o" "t" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, negative StartFraction, sinh left parenthesis, 2 "z" . "x" , right parenthesis Over cos left parenthesis, 2 "z" . "y" , right parenthesis minus cosh left parenthesis, 2 "z" . "x" , right parenthesis , EndFraction , StartFraction, sin left parenthesis, 2 "z" . "y" , right parenthesis Over cos left parenthesis, 2 "z" . "y" , right parenthesis minus cosh left parenthesis, 2 "z" . "x" , right parenthesis , EndFraction , right parenthesisCothz=−sinh2z.xcos2z.y−cosh2z.x,sin2z.ycos2z.y−cosh2z.x
102
𝘐𝘯𝘷𝘦𝘳𝘴𝘦 𝘏𝘺𝘱𝘦𝘳𝘣𝘰𝘭𝘪𝘤 𝘍𝘶𝘯𝘤𝘵𝘪𝘰𝘯𝘴
103
Expression 104: "A" Subscript, "r" "S" "i" "n" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, 0.5ln left parenthesis, left parenthesis, Root Index 4, Start Root, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared plus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared minus 1 , right parenthesis squared , End Root cos left parenthesis, 0.5arctan left parenthesis, 2 "z" . "x" "z" . "y" , "z" . "x" squared minus "z" . "y" squared plus 1 , right parenthesis , right parenthesis plus "z" . "x" , right parenthesis squared plus left parenthesis, Root Index 4, Start Root, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared plus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared minus 1 , right parenthesis squared , End Root sin left parenthesis, 0.5arctan left parenthesis, 2 "z" . "x" "z" . "y" , "z" . "x" squared minus "z" . "y" squared plus 1 , right parenthesis , right parenthesis plus "z" . "y" , right parenthesis squared , right parenthesis , arctan left parenthesis, Root Index 4, Start Root, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared plus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared minus 1 , right parenthesis squared , End Root sin left parenthesis, 0.5arctan left parenthesis, 2 "z" . "x" "z" . "y" , "z" . "x" squared minus "z" . "y" squared plus 1 , right parenthesis , right parenthesis plus "z" . "y" , Root Index 4, Start Root, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared plus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared minus 1 , right parenthesis squared , End Root cos left parenthesis, 0.5arctan left parenthesis, 2 "z" . "x" "z" . "y" , "z" . "x" squared minus "z" . "y" squared plus 1 , right parenthesis , right parenthesis plus "z" . "x" , right parenthesis , right parenthesisArSinhz=0.5ln4z.x4+2z.y2+1z.x2+z.y2−12cos0.5arctan2z.xz.y,z.x2−z.y2+1+z.x2+4z.x4+2z.y2+1z.x2+z.y2−12sin0.5arctan2z.xz.y,z.x2−z.y2+1+z.y2,arctan4z.x4+2z.y2+1z.x2+z.y2−12sin0.5arctan2z.xz.y,z.x2−z.y2+1+z.y,4z.x4+2z.y2+1z.x2+z.y2−12cos0.5arctan2z.xz.y,z.x2−z.y2+1+z.x
104
Expression 105: "A" Subscript, "r" "C" "o" "s" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, 0.5ln left parenthesis, left parenthesis, Root Index 4, Start Root, left parenthesis, left parenthesis, "z" . "x" minus 1 , right parenthesis squared plus "z" . "y" squared , right parenthesis left parenthesis, left parenthesis, "z" . "x" plus 1 , right parenthesis squared plus "z" . "y" squared , right parenthesis , End Root cos left parenthesis, 0.5 left parenthesis, arctan left parenthesis, "z" . "y" , "z" . "x" minus 1 , right parenthesis plus arctan left parenthesis, "z" . "y" , "z" . "x" plus 1 , right parenthesis , right parenthesis , right parenthesis plus "z" . "x" , right parenthesis squared plus left parenthesis, Root Index 4, Start Root, left parenthesis, left parenthesis, "z" . "x" minus 1 , right parenthesis squared plus "z" . "y" squared , right parenthesis left parenthesis, left parenthesis, "z" . "x" plus 1 , right parenthesis squared plus "z" . "y" squared , right parenthesis , End Root sin left parenthesis, 0.5 left parenthesis, arctan left parenthesis, "z" . "y" , "z" . "x" minus 1 , right parenthesis plus arctan left parenthesis, "z" . "y" , "z" . "x" plus 1 , right parenthesis , right parenthesis , right parenthesis plus "z" . "y" , right parenthesis squared , right parenthesis , arctan left parenthesis, Root Index 4, Start Root, left parenthesis, left parenthesis, "z" . "x" minus 1 , right parenthesis squared plus "z" . "y" squared , right parenthesis left parenthesis, left parenthesis, "z" . "x" plus 1 , right parenthesis squared plus "z" . "y" squared , right parenthesis , End Root sin left parenthesis, 0.5 left parenthesis, arctan left parenthesis, "z" . "y" , "z" . "x" minus 1 , right parenthesis plus arctan left parenthesis, "z" . "y" , "z" . "x" plus 1 , right parenthesis , right parenthesis , right parenthesis plus "z" . "y" , Root Index 4, Start Root, left parenthesis, left parenthesis, "z" . "x" minus 1 , right parenthesis squared plus "z" . "y" squared , right parenthesis left parenthesis, left parenthesis, "z" . "x" plus 1 , right parenthesis squared plus "z" . "y" squared , right parenthesis , End Root cos left parenthesis, 0.5 left parenthesis, arctan left parenthesis, "z" . "y" , "z" . "x" minus 1 , right parenthesis plus arctan left parenthesis, "z" . "y" , "z" . "x" plus 1 , right parenthesis , right parenthesis , right parenthesis plus "z" . "x" , right parenthesis , right parenthesisArCoshz=0.5ln4z.x−12+z.y2z.x+12+z.y2cos0.5arctanz.y,z.x−1+arctanz.y,z.x+1+z.x2+4z.x−12+z.y2z.x+12+z.y2sin0.5arctanz.y,z.x−1+arctanz.y,z.x+1+z.y2,arctan4z.x−12+z.y2z.x+12+z.y2sin0.5arctanz.y,z.x−1+arctanz.y,z.x+1+z.y,4z.x−12+z.y2z.x+12+z.y2cos0.5arctanz.y,z.x−1+arctanz.y,z.x+1+z.x
105
Expression 106: "A" Subscript, "r" "T" "a" "n" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, 0.2 5ln left parenthesis, StartFraction, left parenthesis, "z" . "x" plus 1 , right parenthesis squared plus "z" . "y" squared Over left parenthesis, 1 minus "z" . "x" , right parenthesis squared plus "z" . "y" squared , EndFraction , right parenthesis , 0.5 left parenthesis, arctan left parenthesis, "z" . "y" , "z" . "x" plus 1 , right parenthesis minus arctan left parenthesis, negative "z" . "y" , 1 minus "z" . "x" , right parenthesis , right parenthesis , right parenthesisArTanhz=0.25lnz.x+12+z.y21−z.x2+z.y2,0.5arctanz.y,z.x+1−arctan−z.y,1−z.x
106
Expression 107: "A" Subscript, "r" "C" "s" "c" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, 0.5ln left parenthesis, left parenthesis, StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction plus Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared plus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared minus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root cos left parenthesis, 0.5arctan left parenthesis, negative StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "x" squared minus "z" . "y" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis , right parenthesis squared plus left parenthesis, StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction minus Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared plus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared minus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root sin left parenthesis, 0.5arctan left parenthesis, negative StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "x" squared minus "z" . "y" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis , right parenthesis squared , right parenthesis , arctan left parenthesis, Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared plus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared minus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root sin left parenthesis, 0.5arctan left parenthesis, negative StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "x" squared minus "z" . "y" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis minus StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , Root Index 4, Start Root, StartFraction, "z" . "x" to the 4th power plus 2 left parenthesis, "z" . "y" squared plus 1 , right parenthesis "z" . "x" squared plus left parenthesis, "z" . "y" squared minus 1 , right parenthesis squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , End Root cos left parenthesis, 0.5arctan left parenthesis, negative StartFraction, 2 "z" . "x" "z" . "y" Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction , StartFraction, "z" . "x" squared minus "z" . "y" squared Over left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis squared , EndFraction plus 1 , right parenthesis , right parenthesis plus StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesis , right parenthesisArCschz=0.5lnz.xz.x2+z.y2+4z.x4+2z.y2+1z.x2+z.y2−12z.x2+z.y22cos0.5arctan−2z.xz.yz.x2+z.y22,z.x2−z.y2z.x2+z.y22+12+z.yz.x2+z.y2−4z.x4+2z.y2+1z.x2+z.y2−12z.x2+z.y22sin0.5arctan−2z.xz.yz.x2+z.y22,z.x2−z.y2z.x2+z.y22+12,arctan4z.x4+2z.y2+1z.x2+z.y2−12z.x2+z.y22sin0.5arctan−2z.xz.yz.x2+z.y22,z.x2−z.y2z.x2+z.y22+1−z.yz.x2+z.y2,4z.x4+2z.y2+1z.x2+z.y2−12z.x2+z.y22cos0.5arctan−2z.xz.yz.x2+z.y22,z.x2−z.y2z.x2+z.y22+1+z.xz.x2+z.y2
107
Expression 108: "A" Subscript, "r" "S" "e" "c" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, 0.5ln left parenthesis, StartFraction, 2 "z" . "x" plus 2 Over "z" . "x" squared plus "z" . "y" squared , EndFraction Root Index 4, Start Root, StartFraction, "z" . "x" squared minus 2 "z" . "x" plus "z" . "y" squared plus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , End Root cos left parenthesis, 0.5arctan left parenthesis, negative StartFraction, 2 "z" . "y" Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , negative StartFraction, "z" . "x" squared plus "z" . "y" squared minus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , right parenthesis , right parenthesis minus StartFraction, 2 "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction Root Index 4, Start Root, StartFraction, "z" . "x" squared minus 2 "z" . "x" plus "z" . "y" squared plus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , End Root sin left parenthesis, 0.5arctan left parenthesis, negative StartFraction, 2 "z" . "y" Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , negative StartFraction, "z" . "x" squared plus "z" . "y" squared minus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , right parenthesis , right parenthesis plus StartFraction, StartRoot, left parenthesis, "z" . "x" squared minus 2 "z" . "x" plus "z" . "y" squared plus 1 , right parenthesis left parenthesis, left parenthesis, "z" . "x" plus 1 , right parenthesis squared plus "z" . "y" squared , right parenthesis , EndRoot plus 1 Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesis , arctan left parenthesis, StartFraction, left parenthesis, "z" . "x" squared plus "z" . "x" plus "z" . "y" squared , right parenthesis Over "z" . "x" squared plus "z" . "y" squared , EndFraction Root Index 4, Start Root, StartFraction, "z" . "x" squared minus 2 "z" . "x" plus "z" . "y" squared plus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , End Root sin left parenthesis, 0.5arctan left parenthesis, negative StartFraction, 2 "z" . "y" Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , negative StartFraction, "z" . "x" squared plus "z" . "y" squared minus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , right parenthesis , right parenthesis minus StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction left parenthesis, Root Index 4, Start Root, StartFraction, "z" . "x" squared minus 2 "z" . "x" plus "z" . "y" squared plus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , End Root cos left parenthesis, 0.5arctan left parenthesis, negative StartFraction, 2 "z" . "y" Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , negative StartFraction, "z" . "x" squared plus "z" . "y" squared minus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , right parenthesis , right parenthesis plus 1 , right parenthesis , StartFraction, left parenthesis, "z" . "x" squared plus "z" . "x" plus "z" . "y" squared , right parenthesis Over "z" . "x" squared plus "z" . "y" squared , EndFraction Root Index 4, Start Root, StartFraction, "z" . "x" squared minus 2 "z" . "x" plus "z" . "y" squared plus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , End Root cos left parenthesis, 0.5arctan left parenthesis, negative StartFraction, 2 "z" . "y" Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , negative StartFraction, "z" . "x" squared plus "z" . "y" squared minus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , right parenthesis , right parenthesis plus StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction plus StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction Root Index 4, Start Root, StartFraction, "z" . "x" squared minus 2 "z" . "x" plus "z" . "y" squared plus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , End Root sin left parenthesis, 0.5arctan left parenthesis, negative StartFraction, 2 "z" . "y" Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , negative StartFraction, "z" . "x" squared plus "z" . "y" squared minus 1 Over "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , right parenthesis , right parenthesis , right parenthesis , right parenthesisArSechz=0.5ln2z.x+2z.x2+z.y24z.x2−2z.x+z.y2+1z.x2+2z.x+z.y2+1cos0.5arctan−2z.yz.x2+2z.x+z.y2+1,−z.x2+z.y2−1z.x2+2z.x+z.y2+1−2z.yz.x2+z.y24z.x2−2z.x+z.y2+1z.x2+2z.x+z.y2+1sin0.5arctan−2z.yz.x2+2z.x+z.y2+1,−z.x2+z.y2−1z.x2+2z.x+z.y2+1+z.x2−2z.x+z.y2+1z.x+12+z.y2+1z.x2+z.y2,arctanz.x2+z.x+z.y2z.x2+z.y24z.x2−2z.x+z.y2+1z.x2+2z.x+z.y2+1sin0.5arctan−2z.yz.x2+2z.x+z.y2+1,−z.x2+z.y2−1z.x2+2z.x+z.y2+1−z.yz.x2+z.y24z.x2−2z.x+z.y2+1z.x2+2z.x+z.y2+1cos0.5arctan−2z.yz.x2+2z.x+z.y2+1,−z.x2+z.y2−1z.x2+2z.x+z.y2+1+1,z.x2+z.x+z.y2z.x2+z.y24z.x2−2z.x+z.y2+1z.x2+2z.x+z.y2+1cos0.5arctan−2z.yz.x2+2z.x+z.y2+1,−z.x2+z.y2−1z.x2+2z.x+z.y2+1+z.xz.x2+z.y2+z.yz.x2+z.y24z.x2−2z.x+z.y2+1z.x2+2z.x+z.y2+1sin0.5arctan−2z.yz.x2+2z.x+z.y2+1,−z.x2+z.y2−1z.x2+2z.x+z.y2+1
108
Expression 109: "A" Subscript, "r" "C" "o" "t" "h" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, 0.2 5ln left parenthesis, StartFraction, "z" . "x" squared plus 2 "z" . "x" plus "z" . "y" squared plus 1 Over "z" . "x" squared minus 2 "z" . "x" plus "z" . "y" squared plus 1 , EndFraction , right parenthesis , 0.5 left parenthesis, arctan left parenthesis, negative StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction plus 1 , right parenthesis minus arctan left parenthesis, StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , 1 minus StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesis , right parenthesis , right parenthesisArCothz=0.25lnz.x2+2z.x+z.y2+1z.x2−2z.x+z.y2+1,0.5arctan−z.yz.x2+z.y2,z.xz.x2+z.y2+1−arctanz.yz.x2+z.y2,1−z.xz.x2+z.y2
109
𝗭𝗲𝘁𝗮 𝗙𝘂𝗻𝗰𝘁𝗶𝗼𝗻
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110
Expression 111: "Z" Subscript, "e" "t" "a" , Baseline left parenthesis, "z" , right parenthesis equals "D" Subscript, "i" "v" "i" "d" "e" , Baseline left parenthesis, left parenthesis, Start sum from "n" equals 0 to zeta Subscript, "n" , Baseline , end sum, Start sum from "k" equals 0 to "n" , end sum, StartFraction, left parenthesis, negative 1 , right parenthesis Superscript, "k" , Baseline nCr left parenthesis, "n" , "k" , right parenthesis Over 2 Superscript, "n" plus 1 , Baseline , EndFraction left parenthesis, "k" plus 1 , right parenthesis Superscript, negative "z" . "x" , Baseline cos left parenthesis, "z" . "y" ln left parenthesis, "k" plus 1 , right parenthesis , right parenthesis , Start sum from "n" equals 0 to zeta Subscript, "n" , Baseline , end sum, Start sum from "k" equals 0 to "n" , end sum, StartFraction, left parenthesis, negative 1 , right parenthesis Superscript, "k" , Baseline nCr left parenthesis, "n" , "k" , right parenthesis Over 2 Superscript, "n" plus 1 , Baseline , EndFraction left parenthesis, "k" plus 1 , right parenthesis Superscript, negative "z" . "x" , Baseline sin left parenthesis, negative "z" . "y" ln left parenthesis, "k" plus 1 , right parenthesis , right parenthesis , right parenthesis , left parenthesis, negative 2 Superscript, 1 minus "z" . "x" , Baseline cos left parenthesis, "z" . "y" ln left parenthesis, 2 , right parenthesis , right parenthesis plus 1 , 2 Superscript, 1 minus "z" . "x" , Baseline sin left parenthesis, "z" . "y" ln left parenthesis, 2 , right parenthesis , right parenthesis , right parenthesis , right parenthesisZetaz=Divideζn∑n=0n∑k=0−1knCrn,k2n+1k+1−z.xcosz.ylnk+1,ζn∑n=0n∑k=0−1knCrn,k2n+1k+1−z.xsin−z.ylnk+1,−21−z.xcosz.yln2+1,21−z.xsinz.yln2
111
Expression 112: zeta Subscript, "n" , Baseline equals 15ζn=15
112
𝗠𝗶𝘀𝗰𝗲𝗹𝗹𝗮𝗻𝗲𝗼𝘂𝘀 𝗙𝘂𝗻𝗰𝘁𝗶𝗼𝗻𝘀
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113
Expression 114: "R" Subscript, "e" "c" "i" "p" "r" "o" "c" "a" "l" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, StartFraction, "z" . "x" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , negative StartFraction, "z" . "y" Over "z" . "x" squared plus "z" . "y" squared , EndFraction , right parenthesisReciprocalz=z.xz.x2+z.y2,−z.yz.x2+z.y2
114
Expression 115: "S" Subscript, "u" "m" , Baseline left parenthesis, "n" , "m" , "z" , right parenthesis equals left parenthesis, Start sum from "k" equals "n" to "m" , end sum, "z" . "x" , Start sum from "k" equals "n" to "m" , end sum, "z" . "y" , right parenthesisSumn,m,z=m∑k=nz.x,m∑k=nz.y
115
𝗖𝗼𝗺𝗽𝗹𝗲𝘅 𝗖𝗼𝗺𝗽𝗼𝗻𝗲𝗻𝘁𝘀
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116
Expression 117: "A" Subscript, "b" "s" , Baseline left parenthesis, "z" , right parenthesis equals StartRoot, "z" . "x" squared plus "z" . "y" squared , EndRootAbsz=z.x2+z.y2
117
Expression 118: "A" Subscript, "r" "g" , Baseline left parenthesis, "z" , right parenthesis equals arc tangent left parenthesis, "z" . "y" , "z" . "x" , right parenthesisArgz=arctanz.y,z.x
118
Expression 119: "R" Subscript, "e" , Baseline left parenthesis, "z" , right parenthesis equals "z" . "x"Rez=z.x
119
Expression 120: "I" Subscript, "m" , Baseline left parenthesis, "z" , right parenthesis equals "z" . "y"Imz=z.y
120
Expression 121: "C" Subscript, "o" "n" "j" "u" "g" "a" "t" "e" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, "z" . "x" , negative "z" . "y" , right parenthesisConjugatez=z.x,−z.y
121
Expression 122: "S" Subscript, "i" "g" "n" , Baseline left parenthesis, "z" , right parenthesis equals left parenthesis, StartFraction, "z" . "x" Over "A" Subscript, "b" "s" , Baseline left parenthesis, "z" , right parenthesis , EndFraction , StartFraction, "z" . "y" Over "A" Subscript, "b" "s" , Baseline left parenthesis, "z" , right parenthesis , EndFraction , right parenthesisSignz=z.xAbsz,z.yAbsz