CoCalc -- 4338.sagews

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p = complex_plot(lambda z: z^2-1, (-2, 2), (-2, 2))

p.show()

complex_plot(lambda z: log((1+z)/(1-z)), (-2, 2), (-2, 2))

def graph_r(r):
    var('theta,z,u,v,w')
    z = r * e^(i * theta)
    w = log((1 + z)/(1 - z))
    u = log((1 + r^2 + 2*r*cos(theta))/(1 + r^2 - 2*r*cos(theta))) / 2
    v = arctan(sin(theta) * 2*r / (1 - r^2))
    p  = parametric_plot( (theta, u), (theta, -pi, pi), color='blue' )
    p += parametric_plot( (theta, v), (theta, -pi, pi), color='red' )
    p.show()
    err = integrate(abs(u + i*v - w), theta, -pi, pi).n()
    print "error =", err

graph_r(2)

error = 19.7392088021787

var('r,theta,z,u,v,w')
z = r * e^(i * theta)
w = log((1 + z)/(1 - z))
w.real()

\newcommand{\Bold}[1]{\mathbf{#1}}\log\left({\left| -\frac{r e^{\left(i \, \theta\right)} + 1}{r e^{\left(i \, \theta\right)} - 1} \right|}\right)

w.real().simplify()

<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(\frac{\sqrt{r^{2} \sin\left(\theta\right)^{2} + {\left(r \cos\left(\theta\right) + 1\right)}^{2

\sqrt{r^{2} \sin\left(\theta\right)^{2} + {\left(r \cos\left(\theta\right) - 1\right)}^{2}}}\right) }}}

w.real().simplify_full()

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, \log\left(r^{2} - 2 \, r \cos\left(\theta\right) + 1\right) + \frac{1}{2} \, \log\left(r^{2} + 2 \, r \cos\left(\theta\right) + 1\right)

w.imag()

\newcommand{\Bold}[1]{\mathbf{#1}}\arctan\left(\frac{2 \, e^{\left(-\Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right) \Re \left( r \right)}{e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + 2 \, e^{\left(-\Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right) \Im \left( r \right) - 2 \, e^{\left(-\Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right) \Re \left( r \right) + 1} + \frac{2 \, e^{\left(-\Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right) \Im \left( r \right)}{e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + 2 \, e^{\left(-\Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right) \Im \left( r \right) - 2 \, e^{\left(-\Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right) \Re \left( r \right) + 1}, -\frac{e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2}}{e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + 2 \, e^{\left(-\Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right) \Im \left( r \right) - 2 \, e^{\left(-\Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right) \Re \left( r \right) + 1} - \frac{e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2}}{e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + 2 \, e^{\left(-\Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right) \Im \left( r \right) - 2 \, e^{\left(-\Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right) \Re \left( r \right) + 1} - \frac{e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2}}{e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + 2 \, e^{\left(-\Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right) \Im \left( r \right) - 2 \, e^{\left(-\Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right) \Re \left( r \right) + 1} - \frac{e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2}}{e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + 2 \, e^{\left(-\Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right) \Im \left( r \right) - 2 \, e^{\left(-\Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right) \Re \left( r \right) + 1} + \frac{1}{e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Re \left( r \right)^{2} + e^{\left(-2 \, \Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right)^{2} \Im \left( r \right)^{2} + 2 \, e^{\left(-\Im \left( \theta \right)\right)} \sin\left(\Re \left( \theta \right)\right) \Im \left( r \right) - 2 \, e^{\left(-\Im \left( \theta \right)\right)} \cos\left(\Re \left( \theta \right)\right) \Re \left( r \right) + 1}\right)

w.imag().simplify()

\newcommand{\Bold}[1]{\mathbf{#1}}\arctan\left(\frac{2 \, r \sin\left(\theta\right)}{r^{2} \sin\left(\theta\right)^{2} + r^{2} \cos\left(\theta\right)^{2} - 2 \, r \cos\left(\theta\right) + 1}, -\frac{r^{2} \sin\left(\theta\right)^{2}}{r^{2} \sin\left(\theta\right)^{2} + r^{2} \cos\left(\theta\right)^{2} - 2 \, r \cos\left(\theta\right) + 1} - \frac{r^{2} \cos\left(\theta\right)^{2}}{r^{2} \sin\left(\theta\right)^{2} + r^{2} \cos\left(\theta\right)^{2} - 2 \, r \cos\left(\theta\right) + 1} + \frac{1}{r^{2} \sin\left(\theta\right)^{2} + r^{2} \cos\left(\theta\right)^{2} - 2 \, r \cos\left(\theta\right) + 1}\right)

w.imag().simplify_full()

\newcommand{\Bold}[1]{\mathbf{#1}}-\arctan\left(-\frac{2 \, r \sin\left(\theta\right)}{r^{2} - 2 \, r \cos\left(\theta\right) + 1}, -\frac{r^{2} - 1}{r^{2} - 2 \, r \cos\left(\theta\right) + 1}\right)

def gr(r):
    var('theta,z,u,v,v1,v2,w'); assume(theta,'real')
    z = r * e^(i * theta)
    w = log((1 + z)/(1 - z))
    u = log((1 + r^2 + 2*r*cos(theta))/(1 + r^2 - 2*r*cos(theta))) / 2
    v = w.imag()
    v1 = arctan(sin(theta) * 2*r / (1 - r^2)) + pi * sign(theta) * (1 - floor(1/(1 + floor(r))))
    p  = parametric_plot( (theta,  u), (theta, -pi, pi), color='blue' )
    p += parametric_plot( (theta,  v), (theta, -pi, pi), color='black')
    p += parametric_plot( (theta, v1), (theta, -pi, pi), color='red'  )
    p.show()
    #err = numerical_integral(abs(u + i*v1 - w), theta, -pi, pi).n()
    #e0 = numerical_integral(abs(u.imag()),     theta, -pi, pi).n()
    #e1 = numerical_integral(abs(u - w.real()), theta, -pi, pi).n()
    #e2 = numerical_integral(abs(v1- w.imag()), theta, -pi, pi).n()
    err = (sum(abs(u + i*v1 - w).function(theta)(t*pi) for t in range(1,100))/100).simplify_full()
    print "error =", err # there appears to be some error at theta=0
gr(2)

error = 0

gr(3)

error = 1/100*pi

gr(10)

error = 0

gr(1.1)

error = 0.99*I*pi - 1.15463194561e-16 - 3.11017672705*I

gr(1.01)

error = 0.99*I*pi + 1.05426778418e-14 - 3.11017672705*I

gr(1.001)

error = 0.99*I*pi + 6.99991176134e-13 - 3.11017672705*I

gr(.9)

error = 4.39648317752e-16

gr(.7)

error = 3.28626015289e-16

gr(.5)

error = 0

gr(.2)

error = 5.55111512313e-17

gr(.1)

error = 1.24067423002e-16

gr(3)

error = 0

gr(10)

error = 0

var('r,theta,z,u,v,w'); assume(r,'real'); assume(theta,'real')
z = r * e^(i * theta)
w = log((1 + z)/(1 - z))
w.imag().simplify_full()

tan(w.imag()).simplify_full()

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, r \sin\left(\theta\right)}{r^{2} - 1}

e^w.real()*cos(w.imag()).simplify_full()

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(r^{2} - 1\right)} {\left| -\frac{r e^{\left(i \, \theta\right)} + 1}{r e^{\left(i \, \theta\right)} - 1} \right|}}{\sqrt{r^{2} - 2 \, r \cos\left(\theta\right) + 1} \sqrt{r^{2} + 2 \, r \cos\left(\theta\right) + 1}}

((r*e^(i*theta)+1)/(r*e^(i*theta)-1)).simplify_full()

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{r e^{\left(i \, \theta\right)} + 1}{r e^{\left(i \, \theta\right)} - 1}

abs(r*e^(i*theta)+1).simplify_full()

\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{r^{2} + 2 \, r \cos\left(\theta\right) + 1}