Expression 7: "s" Subscript, "e" "l" "f" "p" "o" "w" 2 , Baseline left parenthesis, "z" , right parenthesis equals "e" Superscript, left parenthesis, ln left parenthesis, "r" , right parenthesis plus "i" theta , right parenthesis left parenthesis, "z" . "x" plus "i" "z" . "y" , right parenthesis , Baseline with "r" equals StartAbsoluteValue, "z" , EndAbsoluteValue , theta equals arc tangent left parenthesis, "z" . "y" , "z" . "x" , right parenthesisselfpow2z=elnr+iθz.x+iz.ywithr=z,θ=arctanz.y,z.x
7
Expression 8: "s" Subscript, "e" "l" "f" "p" "o" "w" 3 , Baseline left parenthesis, "z" , right parenthesis equals "e" Superscript, ln left parenthesis, "r" , right parenthesis "z" . "x" minus theta "z" . "y" , Baseline times "e" Superscript, "i" left parenthesis, ln left parenthesis, "r" , right parenthesis "z" . "y" plus theta "z" . "x" , right parenthesis , Baseline with "r" equals StartAbsoluteValue, "z" , EndAbsoluteValue , theta equals arc tangent left parenthesis, "z" . "y" , "z" . "x" , right parenthesisselfpow3z=elnrz.x−θz.y·eilnrz.y+θz.xwithr=z,θ=arctanz.y,z.x
8
Expression 9: "s" Subscript, "e" "l" "f" "p" "o" "w" , Baseline left parenthesis, "z" , right parenthesis equals "e" Superscript, left parenthesis, ln left parenthesis, "r" , right parenthesis "z" . "x" minus theta "z" . "y" , right parenthesis , Baseline times left parenthesis, cos left parenthesis, ln left parenthesis, "r" , right parenthesis "z" . "y" plus theta "z" . "x" , right parenthesis , sin left parenthesis, ln left parenthesis, "r" , right parenthesis "z" . "y" plus theta "z" . "x" , right parenthesis , right parenthesis with "r" equals StartAbsoluteValue, "z" , EndAbsoluteValue , theta equals arc tangent left parenthesis, "z" . "y" , "z" . "x" , right parenthesisselfpowz=elnrz.x−θz.y·coslnrz.y+θz.x,sinlnrz.y+θz.xwithr=z,θ=arctanz.y,z.x
9
Expression 10: "s" left parenthesis, "z" , right parenthesis equals left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis Superscript, StartFraction, "z" . "x" Over 2 , EndFraction , Baseline "e" Superscript, negative "z" . "y" arctan left parenthesis, "z" . "y" , "z" . "x" , right parenthesis , Baseline left parenthesis, cos left parenthesis, "z" . "x" arctan left parenthesis, "z" . "y" , "z" . "x" , right parenthesis plus StartFraction, "z" . "y" Over 2 , EndFraction ln left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis , right parenthesis , sin left parenthesis, "z" . "x" arctan left parenthesis, "z" . "y" , "z" . "x" , right parenthesis plus StartFraction, "z" . "y" Over 2 , EndFraction ln left parenthesis, "z" . "x" squared plus "z" . "y" squared , right parenthesis , right parenthesis , right parenthesissz=z.x2+z.y2z.x2e−z.yarctanz.y,z.xcosz.xarctanz.y,z.x+z.y2lnz.x2+z.y2,sinz.xarctanz.y,z.x+z.y2lnz.x2+z.y2
10
Real part
11
Expression 12: "z" equals "s" left parenthesis, left parenthesis, "x" , "y" , right parenthesis , right parenthesis . "x"z=sx,y.x
12
Imaginary part
13
Expression 14: "z" equals "s" left parenthesis, left parenthesis, "x" , "y" , right parenthesis , right parenthesis . "y"z=sx,y.y
