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Finding the argument of a complex fraction (doing something wrong but not knowing what)
Clash Royale CLAN TAG#URR8PPP
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I am working on a problem, and I think that I miss something very simple but I do not get what I am doing wrong.
The problem:
I need to find the argument of the following function (all the variables are real and greater or equal to zero and I work in radians):
$$argleft(frac1texta_1cdotleft(omegatextjright)^3+texta_2cdotleft(omegatextjright)^2+texta_3cdotleft(omegatextjright)+1right)=$$
$$argleft(frac1-texta_1cdotomega^3cdottextj-texta_2cdotomega^2+texta_3cdotomegacdottextj+1right)=$$
$$argleft(frac11-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)tag1$$
Where $textj^2=i^2=-1$
Using the rules for the agument we can write:
$$argleft(1right)-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)=$$
$$0-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)=$$
$$-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)tag2$$
Setting $argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright):=varphileft(omegaright)$.
My work:
I got the following function for the argument:
$$
varphileft(omegaright)=begincases
-arctanleft(fractexta_3omega-texta_1omega^31-texta_2omega^2right)spacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacetextwhen:spacespace1>texta_2omega^2spacewedgespacetexta_3omega>texta_1omega^3\
\
-left(pi+arctanleft(left|fractexta_3omega-texta_1omega^31-texta_2omega^2right|right)right)spacespacespacespacespacetextwhen:spacespace1<texta_2omega^2spacewedgespacetexta_3omega<texta_1omega^3\
\
-left(fracpi2+arctanleft(left|frac1-texta_2omega^2texta_3omega-texta_1omega^3right|right)right)spacespacespacespacetextwhen:spacespace1<texta_2omega^2spacewedgespacetexta_3omega>texta_1omega^3\
\
-left(frac3pi2+arctanleft(left|frac1-texta_2omega^2texta_3omega-texta_1omega^3right|right)right)spacespacespacetextwhen:spacespace1>texta_2omega^2spacewedgespacetexta_3omega<texta_1omega^3
endcasestag3
$$
But it does not match with the function I got using software, so I'm somewhere wrong in $left(3right)$. And I do not find where
complex-analysis trigonometry complex-numbers vectors conditional-expectation
asked Jul 19 at 18:15
Jan
21.6k31239
 |Â
show 1 more comment
up vote
0
down vote
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I am working on a problem, and I think that I miss something very simple but I do not get what I am doing wrong.
The problem:
I need to find the argument of the following function (all the variables are real and greater or equal to zero and I work in radians):
$$argleft(frac1texta_1cdotleft(omegatextjright)^3+texta_2cdotleft(omegatextjright)^2+texta_3cdotleft(omegatextjright)+1right)=$$
$$argleft(frac1-texta_1cdotomega^3cdottextj-texta_2cdotomega^2+texta_3cdotomegacdottextj+1right)=$$
$$argleft(frac11-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)tag1$$
Where $textj^2=i^2=-1$
Using the rules for the agument we can write:
$$argleft(1right)-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)=$$
$$0-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)=$$
$$-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)tag2$$
Setting $argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright):=varphileft(omegaright)$.
My work:
I got the following function for the argument:
$$
varphileft(omegaright)=begincases
-arctanleft(fractexta_3omega-texta_1omega^31-texta_2omega^2right)spacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacetextwhen:spacespace1>texta_2omega^2spacewedgespacetexta_3omega>texta_1omega^3\
\
-left(pi+arctanleft(left|fractexta_3omega-texta_1omega^31-texta_2omega^2right|right)right)spacespacespacespacespacetextwhen:spacespace1<texta_2omega^2spacewedgespacetexta_3omega<texta_1omega^3\
\
-left(fracpi2+arctanleft(left|frac1-texta_2omega^2texta_3omega-texta_1omega^3right|right)right)spacespacespacespacetextwhen:spacespace1<texta_2omega^2spacewedgespacetexta_3omega>texta_1omega^3\
\
-left(frac3pi2+arctanleft(left|frac1-texta_2omega^2texta_3omega-texta_1omega^3right|right)right)spacespacespacetextwhen:spacespace1>texta_2omega^2spacewedgespacetexta_3omega<texta_1omega^3
endcasestag3
$$
But it does not match with the function I got using software, so I'm somewhere wrong in $left(3right)$. And I do not find where
complex-analysis trigonometry complex-numbers vectors conditional-expectation
asked Jul 19 at 18:15
Jan
21.6k31239
What happened to $a_3$ in your final result?
– Jens
Jul 19 at 18:27
Is it $a_2omega$ or $a_2 omega^2$?
– Jens
Jul 19 at 18:35
@Jens In the final answer I got it right, but not in my problem statement. Now it is correct (thank for your help already!).
– Jan
Jul 19 at 18:37
Have you tried the standard way of defining Atan2?
– Jens
Jul 19 at 19:01
@Jens That is the thing I used.
– Jan
Jul 19 at 19:08
 |Â
show 1 more comment
up vote
0
down vote
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up vote
0
down vote
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I am working on a problem, and I think that I miss something very simple but I do not get what I am doing wrong.
The problem:
I need to find the argument of the following function (all the variables are real and greater or equal to zero and I work in radians):
$$argleft(frac1texta_1cdotleft(omegatextjright)^3+texta_2cdotleft(omegatextjright)^2+texta_3cdotleft(omegatextjright)+1right)=$$
$$argleft(frac1-texta_1cdotomega^3cdottextj-texta_2cdotomega^2+texta_3cdotomegacdottextj+1right)=$$
$$argleft(frac11-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)tag1$$
Where $textj^2=i^2=-1$
Using the rules for the agument we can write:
$$argleft(1right)-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)=$$
$$0-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)=$$
$$-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)tag2$$
Setting $argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright):=varphileft(omegaright)$.
My work:
I got the following function for the argument:
$$
varphileft(omegaright)=begincases
-arctanleft(fractexta_3omega-texta_1omega^31-texta_2omega^2right)spacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacetextwhen:spacespace1>texta_2omega^2spacewedgespacetexta_3omega>texta_1omega^3\
\
-left(pi+arctanleft(left|fractexta_3omega-texta_1omega^31-texta_2omega^2right|right)right)spacespacespacespacespacetextwhen:spacespace1<texta_2omega^2spacewedgespacetexta_3omega<texta_1omega^3\
\
-left(fracpi2+arctanleft(left|frac1-texta_2omega^2texta_3omega-texta_1omega^3right|right)right)spacespacespacespacetextwhen:spacespace1<texta_2omega^2spacewedgespacetexta_3omega>texta_1omega^3\
\
-left(frac3pi2+arctanleft(left|frac1-texta_2omega^2texta_3omega-texta_1omega^3right|right)right)spacespacespacetextwhen:spacespace1>texta_2omega^2spacewedgespacetexta_3omega<texta_1omega^3
endcasestag3
$$
But it does not match with the function I got using software, so I'm somewhere wrong in $left(3right)$. And I do not find where
complex-analysis trigonometry complex-numbers vectors conditional-expectation
asked Jul 19 at 18:15
Jan
21.6k31239
I am working on a problem, and I think that I miss something very simple but I do not get what I am doing wrong.
The problem:
I need to find the argument of the following function (all the variables are real and greater or equal to zero and I work in radians):
$$argleft(frac1texta_1cdotleft(omegatextjright)^3+texta_2cdotleft(omegatextjright)^2+texta_3cdotleft(omegatextjright)+1right)=$$
$$argleft(frac1-texta_1cdotomega^3cdottextj-texta_2cdotomega^2+texta_3cdotomegacdottextj+1right)=$$
$$argleft(frac11-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)tag1$$
Where $textj^2=i^2=-1$
Using the rules for the agument we can write:
$$argleft(1right)-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)=$$
$$0-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)=$$
$$-argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright)tag2$$
Setting $argleft(1-texta_2cdotomega^2+left(texta_3cdotomega-texta_1cdotomega^3right)cdottextjright):=varphileft(omegaright)$.
My work:
I got the following function for the argument:
$$
varphileft(omegaright)=begincases
-arctanleft(fractexta_3omega-texta_1omega^31-texta_2omega^2right)spacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacespacetextwhen:spacespace1>texta_2omega^2spacewedgespacetexta_3omega>texta_1omega^3\
\
-left(pi+arctanleft(left|fractexta_3omega-texta_1omega^31-texta_2omega^2right|right)right)spacespacespacespacespacetextwhen:spacespace1<texta_2omega^2spacewedgespacetexta_3omega<texta_1omega^3\
\
-left(fracpi2+arctanleft(left|frac1-texta_2omega^2texta_3omega-texta_1omega^3right|right)right)spacespacespacespacetextwhen:spacespace1<texta_2omega^2spacewedgespacetexta_3omega>texta_1omega^3\
\
-left(frac3pi2+arctanleft(left|frac1-texta_2omega^2texta_3omega-texta_1omega^3right|right)right)spacespacespacetextwhen:spacespace1>texta_2omega^2spacewedgespacetexta_3omega<texta_1omega^3
endcasestag3
$$
But it does not match with the function I got using software, so I'm somewhere wrong in $left(3right)$. And I do not find where
complex-analysis trigonometry complex-numbers vectors conditional-expectation
asked Jul 19 at 18:15
Jan
21.6k31239
edited Jul 19 at 18:36
asked Jul 19 at 18:15
Jan
21.6k31239
asked Jul 19 at 18:15
Jan
21.6k31239
asked Jul 19 at 18:15
Jan
21.6k31239
21.6k31239
What happened to $a_3$ in your final result?
– Jens
Jul 19 at 18:27
Is it $a_2omega$ or $a_2 omega^2$?
– Jens
Jul 19 at 18:35
@Jens In the final answer I got it right, but not in my problem statement. Now it is correct (thank for your help already!).
– Jan
Jul 19 at 18:37
Have you tried the standard way of defining Atan2?
– Jens
Jul 19 at 19:01
@Jens That is the thing I used.
– Jan
Jul 19 at 19:08
 |Â
show 1 more comment
What happened to $a_3$ in your final result?
– Jens
Jul 19 at 18:27
Is it $a_2omega$ or $a_2 omega^2$?
– Jens
Jul 19 at 18:35
@Jens In the final answer I got it right, but not in my problem statement. Now it is correct (thank for your help already!).
– Jan
Jul 19 at 18:37
Have you tried the standard way of defining Atan2?
– Jens
Jul 19 at 19:01
@Jens That is the thing I used.
– Jan
Jul 19 at 19:08
What happened to $a_3$ in your final result?
– Jens
Jul 19 at 18:27
What happened to $a_3$ in your final result?
– Jens
Jul 19 at 18:27
Is it $a_2omega$ or $a_2 omega^2$?
– Jens
Jul 19 at 18:35
Is it $a_2omega$ or $a_2 omega^2$?
– Jens
Jul 19 at 18:35
@Jens In the final answer I got it right, but not in my problem statement. Now it is correct (thank for your help already!).
– Jan
Jul 19 at 18:37
@Jens In the final answer I got it right, but not in my problem statement. Now it is correct (thank for your help already!).
– Jan
Jul 19 at 18:37
Have you tried the standard way of defining Atan2?
– Jens
Jul 19 at 19:01
Have you tried the standard way of defining Atan2?
– Jens
Jul 19 at 19:01
@Jens That is the thing I used.
– Jan
Jul 19 at 19:08
@Jens That is the thing I used.
– Jan
Jul 19 at 19:08
 |Â
show 1 more comment
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What happened to $a_3$ in your final result?
– Jens
Jul 19 at 18:27
Is it $a_2omega$ or $a_2 omega^2$?
– Jens
Jul 19 at 18:35
@Jens In the final answer I got it right, but not in my problem statement. Now it is correct (thank for your help already!).
– Jan
Jul 19 at 18:37
Have you tried the standard way of defining Atan2?
– Jens
Jul 19 at 19:01
@Jens That is the thing I used.
– Jan
Jul 19 at 19:08