More notes

Define some temporary coordinate systems.

$z-a = (x-a) +iy = {r_1}^{i theta_1} = r_1(cos theta_1 + i sin theta_1)$ , $z+a = (x+a) +iy = {r_2}^{i theta_2} = r_2(cos theta_2 + i sin theta_2)$ , $z-b = (x-b) +iy = {r_3}^{i theta_3} = r_3(cos theta_3 + i sin theta_3)$ , and $z+b = (x+b) +iy = {r_4}^{i theta_4} = r_4(cos theta_4 + i sin theta_4)$ .

⇒ $2z = 2x +2iy = {r_1}^{i theta_1} + {r_2}^{i theta_2}$ ⇒ $z = x +iy = { {r_1}^{i theta_1} + {r_2}^{i theta_2} }/2$ = ${1/2} ( r_1 cos theta_1 + i r_1 sin theta_1 + r_2 cos theta_2 + i r_2 sin theta_2 )$

$Z(z) = {{2P}/pi}{ {z sqrt{a^2 - b^2}} / {(z^2 - b^2)sqrt{z^2-a^2}} }$ = ${{2P}/pi}{ {z sqrt{a^2 - b^2}} / {(z+b)(z-b)sqrt{(z+a)(z-a)}} }$

⇒ $Z(r_1,theta_1,r_2,theta_2,r_3,theta_3,r_4,theta_4) = {{2P}/pi} sqrt{a^2 - b^2} { { {1/2}({r_1}^{i theta_1} + {r_2}^{i theta_2}) } / {{r_3}^{i theta_3}{r_4}^{i theta_4} sqrt{{r_2}^{i theta_2} {r_1}^{i theta_1}}} }$ = ${{P}/pi} sqrt{a^2 - b^2} { { {r_1}^{i theta_1} + {r_2}^{i theta_2} } / {{r_3}^{i theta_3}{r_4}^{i theta_4} {{r_2}^{i theta_2/2} {r_1}^{i theta_1/2}}} }$ = ${{P}/pi} sqrt{a^2 - b^2} { { {r_1}^{i theta_1} + {r_2}^{i theta_2} } /{ {r_1}^{i theta_1/2} {r_2}^{i theta_2/2} } } { { 1 } / { {r_3}^{i theta_3}{r_4}^{i theta_4} } }$ = ${{P}/pi} sqrt{a^2 - b^2} ( { {r_1}^{i theta_1} }{ {r_1}^{-i theta_1/2} {r_2}^{-i theta_2/2} } + { {r_2}^{i theta_2} }{ {r_1}^{-i theta_1/2} {r_2}^{-i theta_2/2} } ) { { 1 } / { {r_3}^{i theta_3}{r_4}^{i theta_4} } }$ = ${{P}/pi} sqrt{a^2 - b^2} { { { {r_1}^{-i theta_1/2} }{ {r_2}^{-i theta_2/2} } + { {r_2}^{-i theta_2/2} }{ {r_1}^{-i theta_1/2} } } / { {r_3}^{i theta_3}{r_4}^{i theta_4} } }$ = ${{2P}/pi} sqrt{a^2 - b^2} { { {r_1}^{-i theta_1/2} {r_2}^{-i theta_2/2} } / { {r_3}^{i theta_3}{r_4}^{i theta_4} } }$ = ${{2P}/pi} sqrt{a^2 - b^2} { {r_1}^{-i theta_1/2} {r_2}^{-i theta_2/2} } { {r_3}^{-i theta_3}{r_4}^{-i theta_4} }$ = ${{2P}/pi} sqrt{a^2 - b^2} (r_1 cos{-theta_1/2} + i r_1 sin{-theta_1/2} ) (r_2 cos{-theta_2/2} + i r_2 sin{-theta_2/2} ) (r_3 cos{-theta_3} + i r_3 sin{-theta_3} ) (r_4 cos{-theta_4} + i r_4 sin{-theta_4} )$ = ${{2P}/pi} sqrt{a^2 - b^2} (r_1 sqrt{{1 + cos theta_1}/2 } - i r_1 sqrt{{1 - cos theta_1}/2 } )$ $( r_2 sqrt{{1 + cos theta_2}/2 } - i r_2 sqrt{{1 - cos theta_2 }/2} ) ( x-b-iy ) ( x+b-iy )$ = ${{2P}/pi} sqrt{a^2 - b^2} (r_1 sqrt{{1 + {{x-a}/r_1}}/2 } - i r_1 sqrt{{1 - {{x-a}/r_1}}/2 } )$ $( r_2 sqrt{{1 + {{x+a}/r_2}}/2 } - i r_2 sqrt{{1 - {{x+a}/r_2}}/2} ) ( x-b-iy ) ( x+b-iy )$ = ${{P}/pi} sqrt{a^2 - b^2} (sqrt{{{r_1}^2 + {r_1(x-a)}} } - i sqrt{{{r_1}^2 - {r_1(x-a)}} } )$ $(sqrt{{{r_2}^2 + {r_2(x+a)}} } - i sqrt{{{r_2}^2 - {r_2(x+a)}}} ) ( x-b-iy ) ( x+b-iy )$ = ${{P}/pi} sqrt{a^2 - b^2} sqrt{r_1 r_2} (sqrt{ r_1 + x-a } - i sqrt{r_1 - x+a } )$ $(sqrt{r_2 + x+a } - i sqrt{r_2 - x-a} ) ( x-b-iy ) ( x+b-iy )$ = ${{P}/pi} sqrt{a^2 - b^2} sqrt{r_1 r_2} (A - i B )(C - i D )( E-iF )( G-iF )$ = ${{P}/pi} sqrt{a^2 - b^2} sqrt{r_1 r_2} (AC -iAD -iBC -BD )( E-iF )( G-iF )$ = ${{P}/pi} sqrt{a^2 - b^2} sqrt{r_1 r_2} (ACE -iACF -iADE -ADF -iBCE -iBCF -BDE +iBDF)( G-iF )$ = ${{P}/pi} sqrt{a^2 - b^2} sqrt{r_1 r_2} (ACEG -iACFG -iADEG -ADFG -iBCEG -iBCFG -BDEG +iBDFG -iACEF -ACF^2 -ADEF +iADF^2 -BCEF -BCF^2 +iBDEF +BDF^2 )$

⇒ $Re[Z]={{P}/pi} sqrt{a^2 - b^2} sqrt{r_1 r_2} (ACEG -ADFG -BDEG -ACF^2 -ADEF -BCEF -BCF^2 +BDF^2 )$ = ${{P}/pi} sqrt{a^2 - b^2} sqrt{r_1 r_2} (AC(x-b)(x+b) -AD y(x+b) -BD(x-b)(x+b) -AC y^2 -AD(x-b) y -BC(x-b) y -BC y^2 +BD y^2 )$

Continue substituting in A, B, C, D, $r_1 = sqrt{(x-a)^2 + y^2}$ , and $r_2 = sqrt{(x+a)^2 + y^2}$ , then simplify that monster, if possible.

Find Z'

$Z(z) = {{2P}/pi}{ {z sqrt{a^2 - b^2}} / {(z^2 - b^2)sqrt{z^2-a^2}} }$

$Z prime(z) = - { {sqrt{a^2-b^2}(2z^4 -a^2z^2 - a^2b^2) }/ {sqrt{z^2-a^2}(z^6 - (2b^2+a^2)z^4 + (b^4+2a^2b^2)z^2 -a^2b^4) } }$ = $- { {sqrt{a^2-b^2}(2z^4 -a^2z^2 - a^2b^2) }/ {sqrt{z^2-a^2}(z^6 - (2b^2+a^2)z^4 + (b^4+2a^2b^2)z^2 -a^2b^4) } }$

Factors:

Numerator: (z^2+ (sqrt(8*a^2*b^2+a^4)-a^2)/4)(z^2 -(sqrt(8*a^2*b^2+a^4)-a^2)/4)

Denominator: (z^2-a^2)(z^2-b^2)

More notes

z*sqrt(a^2 - b^2)/¹⁾

${{2P}/pi} sqrt{a^2 - b^2} (pm sqrt{(x-a)^2 + y^2} sqrt{{1 + cos theta_1}/2 } - pm i sqrt{(x-a)^2 + y^2} sqrt{{1 - cos theta_1}/2 } )$ $(pm sqrt{(x+a)^2 + y^2} sqrt{{1 + cos theta_2}/2 } - pm i sqrt{(x+a)^2 + y^2} sqrt{{1 - cos theta_2 }/2} ) ( x-b-iy ) ( x+b-iy )$ = ${{2P}/pi} sqrt{a^2 - b^2}sqrt{((x-a)^2 + y^2)((x+a)^2 +y^2)} ( sqrt{{1 + cos theta_1}/2 } - i sqrt{{1 - cos theta_1}/2 } )$ $(sqrt{{1 + cos theta_2}/2 } - i sqrt{{1 - cos theta_2 }/2} ) ( x-b-iy ) ( x+b-iy )$

Another Wrong Turn

= ${{2P}/pi} sqrt{a^2 - b^2} (r_1 cos{-theta_1/2} r_2 cos{-theta_2/2} +i r_1 cos{-theta_1/2} r_2 sin{-theta_2/2} +i r_1 sin{-theta_1/2} r_2 cos{-theta_2/2} - r_1 sin{-theta_1/2} r_2 sin{-theta_2/2} )$ $(r_3 cos{-theta_3} r_4 cos{-theta_4} + i r_3 cos{-theta_3} r_4 sin{-theta_4} +i r_3 sin{-theta_3}r_4 cos{-theta_4} - r_3 sin{-theta_3} r_4 sin{-theta_4} )$ = ${{2P}/pi} sqrt{a^2 - b^2} (A + iB + iC -D)(E + iF +iG -H)$ = ${{2P}/pi} sqrt{a^2 - b^2} ( AE +iAF +iAG + AH +iBE - BF - BG -iBH +iCE - CF - CG -iCH - DE -iDF -iDG + DH )$ ⇒ $Re[Z] = {{2P}/pi} sqrt{a^2 - b^2} ( AE + AH - BF - BG - CF - CG - DE + DH )$ = $Re[Z] = {{2P}/pi} sqrt{a^2 - b^2}$ ( $r_1 cos{-theta_1/2} r_2 cos{-theta_2/2} r_3 cos{-theta_3} r_4 cos{-theta_4}$ + $r_1 cos{-theta_1/2} r_2 cos{-theta_2/2} r_3 sin{-theta_3} r_4 sin{-theta_4}$ - $r_1 cos{-theta_1/2} i r_2 sin{-theta_2/2} r_3 cos{-theta_3} r_4 sin{-theta_4}$ - $r_1 cos{-theta_1/2} i r_2 sin{-theta_2/2} r_3 sin{-theta_3}r_4 cos{-theta_4}$ - $r_1 sin{-theta_1/2} r_2 cos{-theta_2/2} r_3 cos{-theta_3} r_4 sin{-theta_4}$ - $r_1 sin{-theta_1/2} r_2 cos{-theta_2/2} r_3 sin{-theta_3}r_4 cos{-theta_4}$ - $r_1 sin{-theta_1/2} r_2 sin{-theta_2/2} r_3 cos{-theta_3} r_4 cos{-theta_4}$ + $r_1 sin{-theta_1/2} r_2 sin{-theta_2/2} r_3 sin{-theta_3} r_4 sin{-theta_4}$ )

Now these need to be transformed back into (x,y).

Was This

Define a couple of temporary coordinate systems.

$z-a = (x-a) +iy = {r_1}^{i theta_1} = r_1(cos theta_1 + i sin theta_1)$ and $z+a = (x+a) +iy = {r_2}^{i theta_2} = r_2(cos theta_2 + i sin theta_2)$ .

$Z(z) = {{2P}/pi}{ {z sqrt{a^2 - b^2}} / {(z^2 - b^2)sqrt{z^2-a^2}} }$ = ${{2P}/pi}{ {z sqrt{a^2 - b^2}} / {(z^2 - b^2)sqrt{(z+a)(z-a)}} }$

⇒ $Z(r_1,theta_1,r_2,theta_2) = {{2P}/pi} sqrt{a^2 - b^2} { { {1/2}({r_1}^{i theta_1} + {r_2}^{i theta_2}) } / {({1/4}({r_1}^{i theta_1} + {r_2}^{i theta_2})^2 - b^2) sqrt{{r_2}^{i theta_2} {r_1}^{i theta_1}}} }$ = ${{P}/pi} sqrt{a^2 - b^2} { { {r_1}^{i theta_1} + {r_2}^{i theta_2} } / {({1/4}({r_1}^{i theta_1} + {r_2}^{i theta_2})^2 - b^2) {{r_2}^{i theta_2/2} {r_1}^{i theta_1/2}}} }$ = ${{P}/pi} sqrt{a^2 - b^2} { { {r_1}^{i theta_1} + {r_2}^{i theta_2} } /{ {r_1}^{i theta_1/2} {r_2}^{i theta_2/2} } } { { 1 } / { {1/4}({r_1}^{i theta_1} + {r_2}^{i theta_2})^2 - b^2 } }$ = ${{P}/pi} sqrt{a^2 - b^2} ( { {r_1}^{i theta_1} }{ {r_1}^{-i theta_1/2} {r_2}^{-i theta_2/2} } + { {r_2}^{i theta_2} }{ {r_1}^{-i theta_1/2} {r_2}^{-i theta_2/2} } ) { { 1 } / { {1/4}({r_1}^{i theta_1} + {r_2}^{i theta_2})^2 - b^2 } }$ = ${{P}/pi} sqrt{a^2 - b^2} { { { {r_1}^{-i theta_1/2} }{ {r_2}^{-i theta_2/2} } + { {r_2}^{-i theta_2/2} }{ {r_1}^{-i theta_1/2} } } / { {1/4}({r_1}^{i theta_1} + {r_2}^{i theta_2})^2 - b^2 } }$ = ${{2P}/pi} sqrt{a^2 - b^2} { { {r_1}^{-i theta_1/2} {r_2}^{-i theta_2/2} } / { {1/4}({r_1}^{i theta_1} + {r_2}^{i theta_2})^2 - b^2 } }$

Didn't Worky

First, change to polar coordinates,

$z = x +iy = r^{i theta} = r(cos theta + i sin theta)$

$Z(z) = {{2P}/pi}{ {z sqrt{a^2 - b^2}} / {(z^2 - b^2)sqrt{z^2-a^2}} }$ = ${{2P}/pi}{ {r^{i theta} sqrt{a^2 - b^2}} / {(r^{i 2 theta} - b^2)sqrt{r^{i 2 theta}-a^2}} }$ = $sqrt{a^2 - b^2} {{2P}/pi} { {r^{i theta} } / {sqrt{ (r^{i 2 theta} - b^2)(r^{i 2 theta} - b^2)(r^{i 2 theta}-a^2)}} }$ = $sqrt{a^2 - b^2} {{2P}/pi} sqrt{ {r^{i 2 theta} } / { (r^{i 2 theta} - b^2)(r^{i 2 theta} - b^2)(r^{i 2 theta}-a^2)} }$

well...

= ${{2P}/pi}{ {sqrt{z^2(a^2 - b^2)}} / {sqrt{(z^2 - b^2)(z^2 - b^2)(z^2-a^2)}} }$ = ${{2P}/pi}{ {sqrt{z^2(a^2 - b^2)}} / {sqrt{(z^2 - b^2)(z^4 -z^2 a^2 - z^2 b^2 +a^4)}} }$ = ${{2P}/pi}{ {sqrt{z^2(a^2 - b^2)}} / {sqrt{(z^2 - b^2)(z^4 +a^4 -z^2(a^2 + b^2) )}} }$ = ${{2P}/pi}{ {sqrt{z^2(a^2 - b^2)}} / {sqrt{z^2 z^2(z^2 - b^2) +a^4(z^2 - b^2) -z^2(z^2 - b^2)(a^2 + b^2) }} }$

hmm

$Z(x+iy) = {{2P}/pi}{ {(x+iy) sqrt{a^2 - b^2}} / {((x+iy)(x+iy) - b^2)sqrt{(x+iy)(x+iy)-a^2}} }$ = $Z(x+iy) = {{2P}/pi}{ {sqrt{(x+iy)(x+iy)(a^2 - b^2)}} / {((x+iy)(x+iy) - b^2)sqrt{(x+iy)(x+iy)-a^2}} }$