How to find all complex solutions?












0












$begingroup$


When i'm solving this Equation
$$z^{2018} + |z|^{1995}* bar{z}^{83} = 2$$



I got this:
$$z_{k} = (cos(frac{2pi k}{1995}) + isin(frac{2pi k}{1995}) * frac{1}{sqrt[2018]{cos(frac{4202pi k}{1995})}}$$
$$k = 0 ... 1994$$
How to find all roots?






calculus trigonometry complex-numbers







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$endgroup$








  • 1




    $begingroup$
    How exactly did you get this? From Wolfram Alpha?
    $endgroup$
    – Kyky
    Dec 9 '18 at 10:32
















0












$begingroup$


When i'm solving this Equation
$$z^{2018} + |z|^{1995}* bar{z}^{83} = 2$$



I got this:
$$z_{k} = (cos(frac{2pi k}{1995}) + isin(frac{2pi k}{1995}) * frac{1}{sqrt[2018]{cos(frac{4202pi k}{1995})}}$$
$$k = 0 ... 1994$$
How to find all roots?






calculus trigonometry complex-numbers







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    How exactly did you get this? From Wolfram Alpha?
    $endgroup$
    – Kyky
    Dec 9 '18 at 10:32














0












0








0





$begingroup$


When i'm solving this Equation
$$z^{2018} + |z|^{1995}* bar{z}^{83} = 2$$



I got this:
$$z_{k} = (cos(frac{2pi k}{1995}) + isin(frac{2pi k}{1995}) * frac{1}{sqrt[2018]{cos(frac{4202pi k}{1995})}}$$
$$k = 0 ... 1994$$
How to find all roots?






calculus trigonometry complex-numbers







share|cite|improve this question









$endgroup$




When i'm solving this Equation
$$z^{2018} + |z|^{1995}* bar{z}^{83} = 2$$



I got this:
$$z_{k} = (cos(frac{2pi k}{1995}) + isin(frac{2pi k}{1995}) * frac{1}{sqrt[2018]{cos(frac{4202pi k}{1995})}}$$
$$k = 0 ... 1994$$
How to find all roots?






calculus trigonometry complex-numbers




calculus trigonometry complex-numbers






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 9 '18 at 10:25









YornYorn

85




85








  • 1




    $begingroup$
    How exactly did you get this? From Wolfram Alpha?
    $endgroup$
    – Kyky
    Dec 9 '18 at 10:32














  • 1




    $begingroup$
    How exactly did you get this? From Wolfram Alpha?
    $endgroup$
    – Kyky
    Dec 9 '18 at 10:32








1




1




$begingroup$
How exactly did you get this? From Wolfram Alpha?
$endgroup$
– Kyky
Dec 9 '18 at 10:32




$begingroup$
How exactly did you get this? From Wolfram Alpha?
$endgroup$
– Kyky
Dec 9 '18 at 10:32










1 Answer
1






active

oldest

votes


















2












$begingroup$

Let $z=re^{it}$



$$implies2=r^{2018}(cos2018t+isin2018t+cos83t-isin83t)$$



Using Prosthaphaeresis Formulas, $$dfrac2{r^{2018}}=2cosdfrac{(2018+83)t}2left(cosdfrac{(2018-83)t}2+isindfrac{(2018-83)t}2right)$$



$impliessindfrac{(2018-83)t}2=0impliesdfrac{(2018-83)t}2=mpi$ where $m$ is any integer



If $m$ is even, $=2n$(say), $impliescosdfrac{(2018-83)t}2=+1$



$$t=dfrac{2npi}{1935},0le2n<1935$$



$$dfrac2{r^{2018}}=2cosdfrac{2101}2cdotdfrac{2npi}{1935}implies r=?$$



What if $m$ is odd, $=2n+1$(say) $impliescosdfrac{(2018-83)t}2=-1$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    what happens if cos(...) = -1?
    $endgroup$
    – Yorn
    Dec 9 '18 at 12:56












  • $begingroup$
    @Yorn, Replace the values of $cos$ & $sin$ of $$dfrac{(2018-83)t}2$$
    $endgroup$
    – lab bhattacharjee
    Dec 9 '18 at 13:07












  • $begingroup$
    $$r = frac{1}{sqrt[2018]{cos(frac{2101pi n}{1995})}}$$ I found r, but I have a problem. I think r should be static
    $endgroup$
    – Yorn
    Dec 9 '18 at 18:07













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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

Let $z=re^{it}$



$$implies2=r^{2018}(cos2018t+isin2018t+cos83t-isin83t)$$



Using Prosthaphaeresis Formulas, $$dfrac2{r^{2018}}=2cosdfrac{(2018+83)t}2left(cosdfrac{(2018-83)t}2+isindfrac{(2018-83)t}2right)$$



$impliessindfrac{(2018-83)t}2=0impliesdfrac{(2018-83)t}2=mpi$ where $m$ is any integer



If $m$ is even, $=2n$(say), $impliescosdfrac{(2018-83)t}2=+1$



$$t=dfrac{2npi}{1935},0le2n<1935$$



$$dfrac2{r^{2018}}=2cosdfrac{2101}2cdotdfrac{2npi}{1935}implies r=?$$



What if $m$ is odd, $=2n+1$(say) $impliescosdfrac{(2018-83)t}2=-1$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    what happens if cos(...) = -1?
    $endgroup$
    – Yorn
    Dec 9 '18 at 12:56












  • $begingroup$
    @Yorn, Replace the values of $cos$ & $sin$ of $$dfrac{(2018-83)t}2$$
    $endgroup$
    – lab bhattacharjee
    Dec 9 '18 at 13:07












  • $begingroup$
    $$r = frac{1}{sqrt[2018]{cos(frac{2101pi n}{1995})}}$$ I found r, but I have a problem. I think r should be static
    $endgroup$
    – Yorn
    Dec 9 '18 at 18:07


















2












$begingroup$

Let $z=re^{it}$



$$implies2=r^{2018}(cos2018t+isin2018t+cos83t-isin83t)$$



Using Prosthaphaeresis Formulas, $$dfrac2{r^{2018}}=2cosdfrac{(2018+83)t}2left(cosdfrac{(2018-83)t}2+isindfrac{(2018-83)t}2right)$$



$impliessindfrac{(2018-83)t}2=0impliesdfrac{(2018-83)t}2=mpi$ where $m$ is any integer



If $m$ is even, $=2n$(say), $impliescosdfrac{(2018-83)t}2=+1$



$$t=dfrac{2npi}{1935},0le2n<1935$$



$$dfrac2{r^{2018}}=2cosdfrac{2101}2cdotdfrac{2npi}{1935}implies r=?$$



What if $m$ is odd, $=2n+1$(say) $impliescosdfrac{(2018-83)t}2=-1$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    what happens if cos(...) = -1?
    $endgroup$
    – Yorn
    Dec 9 '18 at 12:56












  • $begingroup$
    @Yorn, Replace the values of $cos$ & $sin$ of $$dfrac{(2018-83)t}2$$
    $endgroup$
    – lab bhattacharjee
    Dec 9 '18 at 13:07












  • $begingroup$
    $$r = frac{1}{sqrt[2018]{cos(frac{2101pi n}{1995})}}$$ I found r, but I have a problem. I think r should be static
    $endgroup$
    – Yorn
    Dec 9 '18 at 18:07
















2












2








2





$begingroup$

Let $z=re^{it}$



$$implies2=r^{2018}(cos2018t+isin2018t+cos83t-isin83t)$$



Using Prosthaphaeresis Formulas, $$dfrac2{r^{2018}}=2cosdfrac{(2018+83)t}2left(cosdfrac{(2018-83)t}2+isindfrac{(2018-83)t}2right)$$



$impliessindfrac{(2018-83)t}2=0impliesdfrac{(2018-83)t}2=mpi$ where $m$ is any integer



If $m$ is even, $=2n$(say), $impliescosdfrac{(2018-83)t}2=+1$



$$t=dfrac{2npi}{1935},0le2n<1935$$



$$dfrac2{r^{2018}}=2cosdfrac{2101}2cdotdfrac{2npi}{1935}implies r=?$$



What if $m$ is odd, $=2n+1$(say) $impliescosdfrac{(2018-83)t}2=-1$






share|cite|improve this answer









$endgroup$



Let $z=re^{it}$



$$implies2=r^{2018}(cos2018t+isin2018t+cos83t-isin83t)$$



Using Prosthaphaeresis Formulas, $$dfrac2{r^{2018}}=2cosdfrac{(2018+83)t}2left(cosdfrac{(2018-83)t}2+isindfrac{(2018-83)t}2right)$$



$impliessindfrac{(2018-83)t}2=0impliesdfrac{(2018-83)t}2=mpi$ where $m$ is any integer



If $m$ is even, $=2n$(say), $impliescosdfrac{(2018-83)t}2=+1$



$$t=dfrac{2npi}{1935},0le2n<1935$$



$$dfrac2{r^{2018}}=2cosdfrac{2101}2cdotdfrac{2npi}{1935}implies r=?$$



What if $m$ is odd, $=2n+1$(say) $impliescosdfrac{(2018-83)t}2=-1$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 9 '18 at 10:37









lab bhattacharjeelab bhattacharjee

224k15156274




224k15156274












  • $begingroup$
    what happens if cos(...) = -1?
    $endgroup$
    – Yorn
    Dec 9 '18 at 12:56












  • $begingroup$
    @Yorn, Replace the values of $cos$ & $sin$ of $$dfrac{(2018-83)t}2$$
    $endgroup$
    – lab bhattacharjee
    Dec 9 '18 at 13:07












  • $begingroup$
    $$r = frac{1}{sqrt[2018]{cos(frac{2101pi n}{1995})}}$$ I found r, but I have a problem. I think r should be static
    $endgroup$
    – Yorn
    Dec 9 '18 at 18:07




















  • $begingroup$
    what happens if cos(...) = -1?
    $endgroup$
    – Yorn
    Dec 9 '18 at 12:56












  • $begingroup$
    @Yorn, Replace the values of $cos$ & $sin$ of $$dfrac{(2018-83)t}2$$
    $endgroup$
    – lab bhattacharjee
    Dec 9 '18 at 13:07












  • $begingroup$
    $$r = frac{1}{sqrt[2018]{cos(frac{2101pi n}{1995})}}$$ I found r, but I have a problem. I think r should be static
    $endgroup$
    – Yorn
    Dec 9 '18 at 18:07


















$begingroup$
what happens if cos(...) = -1?
$endgroup$
– Yorn
Dec 9 '18 at 12:56






$begingroup$
what happens if cos(...) = -1?
$endgroup$
– Yorn
Dec 9 '18 at 12:56














$begingroup$
@Yorn, Replace the values of $cos$ & $sin$ of $$dfrac{(2018-83)t}2$$
$endgroup$
– lab bhattacharjee
Dec 9 '18 at 13:07






$begingroup$
@Yorn, Replace the values of $cos$ & $sin$ of $$dfrac{(2018-83)t}2$$
$endgroup$
– lab bhattacharjee
Dec 9 '18 at 13:07














$begingroup$
$$r = frac{1}{sqrt[2018]{cos(frac{2101pi n}{1995})}}$$ I found r, but I have a problem. I think r should be static
$endgroup$
– Yorn
Dec 9 '18 at 18:07






$begingroup$
$$r = frac{1}{sqrt[2018]{cos(frac{2101pi n}{1995})}}$$ I found r, but I have a problem. I think r should be static
$endgroup$
– Yorn
Dec 9 '18 at 18:07




















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