= -{P sqrt{b}}/{2 pi}{r_0}^{-2}{r_1}^{-2} ( 3b cos{-1/2 theta_0} cos{-3/2 theta_0} +i 3b sin{-1/2 theta_0} cos{-3/2 theta_0} +i 3b cos{-1/2 theta_0} sin{-3/2 theta_0} -3b sin{-1/2 theta_0} sin{-3/2 theta_0} )(cos{-2 theta_1} +i sin{-2 theta_1})

= -{3 P b sqrt{b}}/{2 pi}{r_0}^{-2}{r_1}^{-2} ( cos{-1/2 theta_0} cos{-3/2 theta_0}cos{-2 theta_1} + i sin{-1/2 theta_0} cos{-3/2 theta_0}cos{-2 theta_1} + i cos{-1/2 theta_0} sin{-3/2 theta_0}cos{-2 theta_1} - sin{-1/2 theta_0} sin{-3/2 theta_0}cos{-2 theta_1} +i cos{-1/2 theta_0} cos{-3/2 theta_0}sin{-2 theta_1} - sin{-1/2 theta_0} cos{-3/2 theta_0}sin{-2 theta_1} - cos{-1/2 theta_0} sin{-3/2 theta_0}sin{-2 theta_1} - isin{-1/2 theta_0} sin{-3/2 theta_0}sin{-2 theta_1} )

Im Z' = -{3Pb sqrt{b}}/{2 pi}{r_0}^{-2}{r_1}^{-2} ( sin{-1/2 theta_0} cos{-3/2 theta_0}cos{-2 theta_1} +cos{-1/2 theta_0} sin{-3/2 theta_0}cos{-2 theta_1} +cos{-1/2 theta_0} cos{-3/2 theta_0}sin{-2 theta_1} -sin{-1/2 theta_0} sin{-3/2 theta_0}sin{-2 theta_1} )

y=r_0 sin theta_0⇒

y Im Z' = -{3Pb sqrt{b}}/{2 pi}{r_0}^{-1}{r_1}^{-2} ( sin{-1/2 theta_0} cos{-3/2 theta_0}cos{-2 theta_1}sin theta_0 +cos{-1/2 theta_0} sin{-3/2 theta_0}cos{-2 theta_1}sin theta_0 +cos{-1/2 theta_0} cos{-3/2 theta_0}sin{-2 theta_1}sin theta_0 -sin{-1/2 theta_0} sin{-3/2 theta_0}sin{-2 theta_1}sin theta_0 )

= -{3Pb sqrt{b}}/{2 pi}{r_0}^{-1}{r_1}^{-2} ( -sin{1/2 theta_0} cos{3/2 theta_0}cos{2 theta_1}sin theta_0 -cos{1/2 theta_0} sin{3/2 theta_0}cos{2 theta_1}sin theta_0 -cos{1/2 theta_0} cos{3/2 theta_0}sin{2 theta_1}sin theta_0 +sin{1/2 theta_0} sin{3/2 theta_0}sin{2 theta_1}sin theta_0 )

= {3Pb sqrt{b}}/{2 pi}{r_0}^{-1}{r_1}^{-2} ( sin{1/2 theta_0} cos{3/2 theta_0}cos{2 theta_1}sin theta_0 +cos{1/2 theta_0} sin{3/2 theta_0}cos{2 theta_1}sin theta_0 +cos{1/2 theta_0} cos{3/2 theta_0}sin{2 theta_1}sin theta_0 -sin{1/2 theta_0} sin{3/2 theta_0}sin{2 theta_1}sin theta_0 )

= {3Pb sqrt{b}}/{2 pi}{r_0}^{-1}{r_1}^{-2} sin theta_0 ( sin{1/2 theta_0} cos{3/2 theta_0}cos{2 theta_1} +cos{1/2 theta_0} sin{3/2 theta_0}cos{2 theta_1} +cos{1/2 theta_0} cos{3/2 theta_0}sin{2 theta_1} -sin{1/2 theta_0} sin{3/2 theta_0}sin{2 theta_1} )

Im Z(r_0,theta_0,r_1,theta_1) = {P sqrt{b}}/{pi} {1/{r_1 sqrt{r_0}}} ( cos{-theta_1}sin{-1/2}theta_0 + sin{-theta_1}cos{-1/2}theta_0 ) = Im Z(r_0,theta_0,r_1,theta_1) = -{P sqrt{b}}/{pi} {1/{r_1 sqrt{r_0}}} ( cos{theta_1}sin{1/2}theta_0 + sin{theta_1}cos{1/2}theta_0 ).

⇒ Im Z(x,y,r_0,r_1) = -{P sqrt{b}}/{pi} {1/{r_1 sqrt{r_0}}} ( {{x+b}/{r_1}} {1/sqrt{2}} sqrt{1-{x/r_0}} + {{y}/{r_1}} {1/sqrt{2}} sqrt{1+{x/r_0}} ) = -{P sqrt{b}}/{pi sqrt{2}} {1/{{r_1}^2 sqrt{r_0}}} ( (x+b) sqrt{1-{x/r_0}} + y sqrt{1+{x/r_0}} ) = -{P sqrt{b}}/{pi sqrt{2}} {1/{{r_1}^2 sqrt{r_0}}} ( (x+b) sqrt{{r_0-x}/r_0} + y sqrt{{r_0+x}/r_0} ) = -{P sqrt{b}}/{pi sqrt{2}} {1/{{r_1}^2 r_0}} ( (x+b) sqrt{r_0-x} + y sqrt{r_0+x} ) ⇒ Im Z(x,y) = -{P sqrt{b}}/{pi sqrt{2}} 1/{((x+b)^2+y^2) sqrt{x^2+y^2}} ( (x+b) sqrt{sqrt{x^2+y^2}-x} + y sqrt{sqrt{x^2+y^2}+x} ) = Im Z(x,y) = -{P sqrt{b}}/{pi sqrt{2}} 1/{((x+b)^2+y^2) sqrt{x^2+y^2}} ( (x+b) sqrt{sqrt{(x+iy)(x-iy)}-x} + y sqrt{sqrt{(x+iy)(x-iy)}+x} )

Im Z' = -{{partial Re Z(z)}/{partial y}}

  • sin{theta_0/2} = pm sqrt{{1-cos theta_0}/2} = pm {1/sqrt{2}}sqrt{1-{x/r_0}}, and
  • cos{theta_0/2} = pm sqrt{{1+cos theta_0}/2} = pm {1/sqrt{2}}sqrt{1+{x/r_0}}, gives

⇒ Re Z(x,y,r_0,r_1) = {P sqrt{b}}/{pi} 1/{ r_1 sqrt{r_0} } ( {x+b}/{r_1} {1/sqrt{2}}sqrt{1+{x/r_0}} - y/{r_1} {1/sqrt{2}}sqrt{1-{x/r_0}} ) = {P sqrt{b}}/{pi sqrt{2}} 1/{ {r_1}^2 sqrt{r_0} } ( (x+b) sqrt{1+{x/r_0}} - y sqrt{1-{x/r_0}} ) = {P sqrt{b}}/{pi sqrt{2}} 1/{ {r_1}^2 sqrt{r_0} } ( (x+b) sqrt{{r_0+x}/r_0} - y sqrt{{r_0-x}/r_0} ) = {P sqrt{b}}/{pi sqrt{2}} 1/{ {r_1}^2 r_0} ( (x+b) sqrt{r_0+x} - y sqrt{r_0-x} ).

⇒ Re Z(x,y) = {P sqrt{b}}/{pi sqrt{2}} 1/{ ((x+b)^2+y^2) sqrt{x^2+y^2}} ( (x+b) sqrt{sqrt{x^2+y^2}+x} - y sqrt{sqrt{x^2+y^2}-x} ) = Re Z(x,y) = {P sqrt{b}}/{pi sqrt{2}} 1/{ (x^2 +2bx +b^2+y^2) sqrt{(x+iy)(x-iy)}} ( (x+b) sqrt{sqrt{(x+iy)(x-iy)}+x} - y sqrt{sqrt{(x+iy)(x-iy)}-x} ) = Re Z(x,y) = {P sqrt{b}}/{pi sqrt{2}} 1/{ (x+b+iy)(x+b-iy) sqrt{z(x-iy)}} ( (x+b) sqrt{sqrt{z(x-iy)}+x} - y sqrt{sqrt{z(x-iy)}-x} ) = Re Z(x,y) = {P sqrt{b}}/{pi sqrt{2}} 1/{ (z+b)(x+b-iy) sqrt{z(x-iy)}} ( (x+b) sqrt{sqrt{z(x-iy)}+x} - y sqrt{sqrt{z(x-iy)}-x} )

= {P sqrt{b}}/{pi sqrt{2}} 1/{ sqrt{((x+b)^2+y^2)^2 (x^2+y^2)}} ( sqrt{(x+b)^2 sqrt{x^2+y^2}+x(x+b)^2} - sqrt{y^2 sqrt{x^2+y^2}-xy^2} ) = {P sqrt{b}}/{pi sqrt{2}} 1/{ sqrt{((x+b)^2+y^2)^2 (x^2+y^2)}} ( sqrt{ sqrt{(x+b)^4 (x^2+y^2)}+x(x+b)^2} - sqrt{sqrt{y^4 (x^2+y^2)}-xy^2} )

⇒ Z(x,y,theta_0) = {P sqrt{b}}/{pi} 1/{ sqrt{(x+b)^2+y^2} root{4}{x^2+y^2} } ( {x+b}/{sqrt{(x+b)^2+y^2}} cos{1/2}theta_0 - y/{sqrt{(x+b)^2+y^2}} sin{1/2}theta_0 ).

  • sin{theta_0/2} = pm sqrt{{1-cos theta_0}/2} = pm {1/sqrt{2}}sqrt{1-{x/sqrt{x^2+y^2}}}, and
  • cos{theta_0/2} = pm sqrt{{1+cos theta_0}/2} = pm {1/sqrt{2}}sqrt{1+{x/sqrt{x^2+y^2}}}, gives

Z(x,y,theta_0) = {P sqrt{b}}/{pi} 1/{ sqrt{(x+b)^2+y^2} root{4}{x^2+y^2} } ( {x+b}/{sqrt{(x+b)^2+y^2}} cos{1/2}theta_0 - y/{sqrt{(x+b)^2+y^2}} sin{1/2}theta_0 ).

Z'(z) = -{P}/{pi} sqrt{{b}/{z}} {3z+b}/{2z(z+b)^2} = -{P sqrt{b}}/{2 pi} z^{-3/2}(3z+b)(z+b)^{-2} = -{P sqrt{b}}/{2 pi} {r_0}^{-3/2} e^{-3/2 theta_0 i} (3 r_0 e^{theta_0 i} +b) {r_1}^{-2} e^{-2 theta_1 i} = -{P sqrt{b}}/{2 pi} (3 {r_0}^{-5/2} e^{-5/2 theta_0 i} +b {r_0}^{-3/2} e^{-3/2 theta_0 i}) {r_1}^{-2} e^{-2 theta_1 i} = -{P sqrt{b}}/{2 pi} (3 {r_0}^{-5/2} (cos{-5/2 theta_0} + i sin{-5/2 theta_0}) +b {r_0}^{-3/2} (cos{-3/2 theta_0} + i sin{-3/2 theta_0})) {r_1}^{-2} (cos{-2 theta_1} +i sin{-2 theta_1})

-OR-

  • 3z+b = (3x+b) +i3y = r_2 e^{i theta_2} = r_2(cos theta_2 + i sin theta_2)

= -{P sqrt{b}}/{2 pi} {r_0}^{-3/2} e^{-3/2 theta_0 i} r_2 e^{i theta_2} {r_1}^{-2} e^{-2 theta_1 i}

1/(z+b)*sqrt(b/z)

dZ/dz = -sqrt(b)*(3*z+b)/(sqrt(z)*(2*z^3+4*b*z^2+2*b^2*z)) = -sqrt{{b}/{z}} {3z+b}/{2z(z^2 + 2bz +b^2)} = -sqrt{{b}/{z}} {3z+b}/{2z(z+b)^2}

-{P sqrt{b}}/{2 pi} {3r_0 e^{i theta_0}+b}/{r_0 e^{i {3/2}theta_0} r_1 e^{i 2 theta_1}} =

=

= {P sqrt{b}}/{pi} sqrt{1/{z(z+b)^2}} = {P sqrt{b}}/{pi} sqrt{1/{r_0 e^{i theta_0} r_1 e^{i 2 theta_1}}} = {P sqrt{b}}/{pi} 1/sqrt{r_0 r_1} e^{-i{1/2} theta_0} e^{ -i theta_1} = {P sqrt{b}}/{pi sqrt{r_0 r_1}} e^{i({-1/2} theta_0 -theta_1)} = {P sqrt{b}}/{pi sqrt{r_0 r_1}} cos({-1/2} theta_0 -theta_1) + i sin({-1/2} theta_0 -theta_1)

{Pb}/{pi sqrt{r_0 r_1}} 1/{ (cos {1/2}theta_0 + i sin {1/2}theta_0) (cos theta_1 + i sin theta_1) } = {Pb}/{pi sqrt{r_0 r_1}} {1/{ (cos {1/2}theta_0 + i sin {1/2}theta_0) (cos theta_1 + i sin theta_1) } } { (cos {1/2}theta_0 - i sin {1/2}theta_0) (cos theta_1 - i sin theta_1) } /{(cos {1/2}theta_0 - i sin {1/2}theta_0) (cos theta_1 - i sin theta_1) } = {Pb}/{pi sqrt{r_0 r_1}} {{cos {1/2}theta_0 cos theta_1 +sin {1/2}theta_0 sin theta_1 -i(sin {1/2}theta_0 cos theta_1+cos {1/2}theta_0 sin theta_1)}/{ (cos^2 {1/2}theta_0 + sin^2 {1/2}theta_0) (cos^2 theta_1 + sin^2 theta_1) } }

{Pb}/{pi} sqrt{1/{r_0 r_1 e^{i (theta_0 + 2 theta_1)}}} = {Pb}/{pi} (r_0 r_1 e^{i (theta_0 + 2 theta_1)})^{-1/2} = {Pb}/{pi} 1/sqrt{r_0 r_1} e^{i (-theta_0/2 -theta_1)}

z = x+iy ⇒ Z(x,y) = {P}/{pi} {1}/{x +iy +b} sqrt{b/{x + iy}}

sigma_y = ReZ + y(Im Z prime + Im Y prime)


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