Find the different principal values.

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Show that the principal value of $i^{i^i}$ differs from the principal value of $i^{i*i}$.
And find the set of all values of the expression $i^{i^i}$.
complex-analysis
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Show that the principal value of $i^{i^i}$ differs from the principal value of $i^{i*i}$.
And find the set of all values of the expression $i^{i^i}$.
complex-analysis
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Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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up vote
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down vote
favorite
up vote
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down vote
favorite
Show that the principal value of $i^{i^i}$ differs from the principal value of $i^{i*i}$.
And find the set of all values of the expression $i^{i^i}$.
complex-analysis
New contributor
Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Show that the principal value of $i^{i^i}$ differs from the principal value of $i^{i*i}$.
And find the set of all values of the expression $i^{i^i}$.
complex-analysis
complex-analysis
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Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Peter van de Berg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked Nov 21 at 12:27
Peter van de Berg
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168
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1 Answer
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First, $i^{i*i}=i^{-1}=frac{1}{i}=-i$.
Then, $i^{i^i}=i^{(e^{frac{pi}{2}i+2pi mi})^i}=i^{e^{-frac{pi}{2}-2pi m}}=(cis(frac{pi}{2}))^{(e^{-frac{pi}{2}-2pi m})}=cis(frac{pi}{2}e^{-frac{pi}{2}-2pi m})$, using the notation $cis(x)=cos(x)+i*sin(x)$, and for some integer $m$. The principal value will be for $m=0$, so $i^{i^i}=cis(frac{pi}{2}e^{-frac{pi}{2}})$.
In order to show that there is no value for $m$ for which the two expressions are equal, we will suppose that the value $m=n$ will fulfill $cis(frac{pi}{2}e^{-frac{pi}{2}-2pi m})=-i$
$$-i=cis(frac{3pi}{2})=cis(frac{pi}{2}e^{-frac{pi}{2}-2pi n})$$
$$frac{3pi}{2}=frac{pi}{2} e^{-frac{pi}{2}-2pi n} $$
$$3=e^{-frac{pi}{2}-2pi n}$$
$$frac{1}{3}=e^{frac{pi}{2}+2pi n}$$
$$n=-frac{frac{pi}{2}+ln(3)}{2pi}=-frac{1}{4}-frac{ln(3)}{2pi}$$
As $frac{ln(3)}{2pi}$ is irrational, $n$ must be irrational, and as such, cannot be equal to any integer $m$.
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Thankyou so much! This will help me
– Peter van de Berg
Nov 22 at 12:21
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
First, $i^{i*i}=i^{-1}=frac{1}{i}=-i$.
Then, $i^{i^i}=i^{(e^{frac{pi}{2}i+2pi mi})^i}=i^{e^{-frac{pi}{2}-2pi m}}=(cis(frac{pi}{2}))^{(e^{-frac{pi}{2}-2pi m})}=cis(frac{pi}{2}e^{-frac{pi}{2}-2pi m})$, using the notation $cis(x)=cos(x)+i*sin(x)$, and for some integer $m$. The principal value will be for $m=0$, so $i^{i^i}=cis(frac{pi}{2}e^{-frac{pi}{2}})$.
In order to show that there is no value for $m$ for which the two expressions are equal, we will suppose that the value $m=n$ will fulfill $cis(frac{pi}{2}e^{-frac{pi}{2}-2pi m})=-i$
$$-i=cis(frac{3pi}{2})=cis(frac{pi}{2}e^{-frac{pi}{2}-2pi n})$$
$$frac{3pi}{2}=frac{pi}{2} e^{-frac{pi}{2}-2pi n} $$
$$3=e^{-frac{pi}{2}-2pi n}$$
$$frac{1}{3}=e^{frac{pi}{2}+2pi n}$$
$$n=-frac{frac{pi}{2}+ln(3)}{2pi}=-frac{1}{4}-frac{ln(3)}{2pi}$$
As $frac{ln(3)}{2pi}$ is irrational, $n$ must be irrational, and as such, cannot be equal to any integer $m$.
New contributor
MoKo19 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thankyou so much! This will help me
– Peter van de Berg
Nov 22 at 12:21
add a comment |
up vote
0
down vote
accepted
First, $i^{i*i}=i^{-1}=frac{1}{i}=-i$.
Then, $i^{i^i}=i^{(e^{frac{pi}{2}i+2pi mi})^i}=i^{e^{-frac{pi}{2}-2pi m}}=(cis(frac{pi}{2}))^{(e^{-frac{pi}{2}-2pi m})}=cis(frac{pi}{2}e^{-frac{pi}{2}-2pi m})$, using the notation $cis(x)=cos(x)+i*sin(x)$, and for some integer $m$. The principal value will be for $m=0$, so $i^{i^i}=cis(frac{pi}{2}e^{-frac{pi}{2}})$.
In order to show that there is no value for $m$ for which the two expressions are equal, we will suppose that the value $m=n$ will fulfill $cis(frac{pi}{2}e^{-frac{pi}{2}-2pi m})=-i$
$$-i=cis(frac{3pi}{2})=cis(frac{pi}{2}e^{-frac{pi}{2}-2pi n})$$
$$frac{3pi}{2}=frac{pi}{2} e^{-frac{pi}{2}-2pi n} $$
$$3=e^{-frac{pi}{2}-2pi n}$$
$$frac{1}{3}=e^{frac{pi}{2}+2pi n}$$
$$n=-frac{frac{pi}{2}+ln(3)}{2pi}=-frac{1}{4}-frac{ln(3)}{2pi}$$
As $frac{ln(3)}{2pi}$ is irrational, $n$ must be irrational, and as such, cannot be equal to any integer $m$.
New contributor
MoKo19 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Thankyou so much! This will help me
– Peter van de Berg
Nov 22 at 12:21
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
First, $i^{i*i}=i^{-1}=frac{1}{i}=-i$.
Then, $i^{i^i}=i^{(e^{frac{pi}{2}i+2pi mi})^i}=i^{e^{-frac{pi}{2}-2pi m}}=(cis(frac{pi}{2}))^{(e^{-frac{pi}{2}-2pi m})}=cis(frac{pi}{2}e^{-frac{pi}{2}-2pi m})$, using the notation $cis(x)=cos(x)+i*sin(x)$, and for some integer $m$. The principal value will be for $m=0$, so $i^{i^i}=cis(frac{pi}{2}e^{-frac{pi}{2}})$.
In order to show that there is no value for $m$ for which the two expressions are equal, we will suppose that the value $m=n$ will fulfill $cis(frac{pi}{2}e^{-frac{pi}{2}-2pi m})=-i$
$$-i=cis(frac{3pi}{2})=cis(frac{pi}{2}e^{-frac{pi}{2}-2pi n})$$
$$frac{3pi}{2}=frac{pi}{2} e^{-frac{pi}{2}-2pi n} $$
$$3=e^{-frac{pi}{2}-2pi n}$$
$$frac{1}{3}=e^{frac{pi}{2}+2pi n}$$
$$n=-frac{frac{pi}{2}+ln(3)}{2pi}=-frac{1}{4}-frac{ln(3)}{2pi}$$
As $frac{ln(3)}{2pi}$ is irrational, $n$ must be irrational, and as such, cannot be equal to any integer $m$.
New contributor
MoKo19 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
First, $i^{i*i}=i^{-1}=frac{1}{i}=-i$.
Then, $i^{i^i}=i^{(e^{frac{pi}{2}i+2pi mi})^i}=i^{e^{-frac{pi}{2}-2pi m}}=(cis(frac{pi}{2}))^{(e^{-frac{pi}{2}-2pi m})}=cis(frac{pi}{2}e^{-frac{pi}{2}-2pi m})$, using the notation $cis(x)=cos(x)+i*sin(x)$, and for some integer $m$. The principal value will be for $m=0$, so $i^{i^i}=cis(frac{pi}{2}e^{-frac{pi}{2}})$.
In order to show that there is no value for $m$ for which the two expressions are equal, we will suppose that the value $m=n$ will fulfill $cis(frac{pi}{2}e^{-frac{pi}{2}-2pi m})=-i$
$$-i=cis(frac{3pi}{2})=cis(frac{pi}{2}e^{-frac{pi}{2}-2pi n})$$
$$frac{3pi}{2}=frac{pi}{2} e^{-frac{pi}{2}-2pi n} $$
$$3=e^{-frac{pi}{2}-2pi n}$$
$$frac{1}{3}=e^{frac{pi}{2}+2pi n}$$
$$n=-frac{frac{pi}{2}+ln(3)}{2pi}=-frac{1}{4}-frac{ln(3)}{2pi}$$
As $frac{ln(3)}{2pi}$ is irrational, $n$ must be irrational, and as such, cannot be equal to any integer $m$.
New contributor
MoKo19 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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answered Nov 21 at 13:08
MoKo19
813
813
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MoKo19 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thankyou so much! This will help me
– Peter van de Berg
Nov 22 at 12:21
add a comment |
Thankyou so much! This will help me
– Peter van de Berg
Nov 22 at 12:21
Thankyou so much! This will help me
– Peter van de Berg
Nov 22 at 12:21
Thankyou so much! This will help me
– Peter van de Berg
Nov 22 at 12:21
add a comment |
Peter van de Berg is a new contributor. Be nice, and check out our Code of Conduct.
Peter van de Berg is a new contributor. Be nice, and check out our Code of Conduct.
Peter van de Berg is a new contributor. Be nice, and check out our Code of Conduct.
Peter van de Berg is a new contributor. Be nice, and check out our Code of Conduct.
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