Can we write complex functions in polar form? Consider the cases $mathbb Rto mathbb C$ and $mathbb Cto mathbb...
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For a complex number $zin mathbb C$, we can write $z=a+ib=sqrt{a^2+b^2}, e^{iphi}$, where $phi=arctanfrac{b}{a}$ and $a,b,thetain mathbb R$ .
Is this also true for complex functions?
I'm interested in two cases.
Case 1:
Let $f:mathbb Rtomathbb C$ and $u,v:mathbb Rto mathbb R$.
Can we write
$$f(t)=u(t)+iv(t)=sqrt{u(t)^2+v(t)^2}, e^{itheta (t)},$$
where $theta:mathbb Rto mathbb R$, so $theta (t)=arctan{frac{v(t)}{u(t)}}$ ?
Case 2:
Let $f:mathbb Ctomathbb C$ and $u,v:mathbb R^2to mathbb R$.
Can we write
begin{align}
f(z)&=f(x+iy)=u(x,y)+iv(x,y)\
&=sqrt{u(x,y)^2+v(x,y)^2}, e^{itheta (x,y)},
end{align}
where $theta:mathbb R^2to mathbb R$, so $theta (x,y)=arctan{frac{v(x,y)}{u(x,y)}}$ ?
Are these cases correct?
complex-analysis
asked Nov 14 at 10:23
JDoeDoe
7711513
add a comment |
up vote
0
down vote
favorite
For a complex number $zin mathbb C$, we can write $z=a+ib=sqrt{a^2+b^2}, e^{iphi}$, where $phi=arctanfrac{b}{a}$ and $a,b,thetain mathbb R$ .
Is this also true for complex functions?
I'm interested in two cases.
Case 1:
Let $f:mathbb Rtomathbb C$ and $u,v:mathbb Rto mathbb R$.
Can we write
$$f(t)=u(t)+iv(t)=sqrt{u(t)^2+v(t)^2}, e^{itheta (t)},$$
where $theta:mathbb Rto mathbb R$, so $theta (t)=arctan{frac{v(t)}{u(t)}}$ ?
Case 2:
Let $f:mathbb Ctomathbb C$ and $u,v:mathbb R^2to mathbb R$.
Can we write
begin{align}
f(z)&=f(x+iy)=u(x,y)+iv(x,y)\
&=sqrt{u(x,y)^2+v(x,y)^2}, e^{itheta (x,y)},
end{align}
where $theta:mathbb R^2to mathbb R$, so $theta (x,y)=arctan{frac{v(x,y)}{u(x,y)}}$ ?
Are these cases correct?
complex-analysis
asked Nov 14 at 10:23
JDoeDoe
7711513
$arctanfrac{v(t)}{u(t)}$ might not be defined, $u$ might have zeros, the definition of $theta$ is not that lucky, better use the argument function.
– Peter Melech
Nov 14 at 11:16
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
For a complex number $zin mathbb C$, we can write $z=a+ib=sqrt{a^2+b^2}, e^{iphi}$, where $phi=arctanfrac{b}{a}$ and $a,b,thetain mathbb R$ .
Is this also true for complex functions?
I'm interested in two cases.
Case 1:
Let $f:mathbb Rtomathbb C$ and $u,v:mathbb Rto mathbb R$.
Can we write
$$f(t)=u(t)+iv(t)=sqrt{u(t)^2+v(t)^2}, e^{itheta (t)},$$
where $theta:mathbb Rto mathbb R$, so $theta (t)=arctan{frac{v(t)}{u(t)}}$ ?
Case 2:
Let $f:mathbb Ctomathbb C$ and $u,v:mathbb R^2to mathbb R$.
Can we write
begin{align}
f(z)&=f(x+iy)=u(x,y)+iv(x,y)\
&=sqrt{u(x,y)^2+v(x,y)^2}, e^{itheta (x,y)},
end{align}
where $theta:mathbb R^2to mathbb R$, so $theta (x,y)=arctan{frac{v(x,y)}{u(x,y)}}$ ?
Are these cases correct?
complex-analysis
asked Nov 14 at 10:23
JDoeDoe
7711513
For a complex number $zin mathbb C$, we can write $z=a+ib=sqrt{a^2+b^2}, e^{iphi}$, where $phi=arctanfrac{b}{a}$ and $a,b,thetain mathbb R$ .
Is this also true for complex functions?
I'm interested in two cases.
Case 1:
Let $f:mathbb Rtomathbb C$ and $u,v:mathbb Rto mathbb R$.
Can we write
$$f(t)=u(t)+iv(t)=sqrt{u(t)^2+v(t)^2}, e^{itheta (t)},$$
where $theta:mathbb Rto mathbb R$, so $theta (t)=arctan{frac{v(t)}{u(t)}}$ ?
Case 2:
Let $f:mathbb Ctomathbb C$ and $u,v:mathbb R^2to mathbb R$.
Can we write
begin{align}
f(z)&=f(x+iy)=u(x,y)+iv(x,y)\
&=sqrt{u(x,y)^2+v(x,y)^2}, e^{itheta (x,y)},
end{align}
where $theta:mathbb R^2to mathbb R$, so $theta (x,y)=arctan{frac{v(x,y)}{u(x,y)}}$ ?
Are these cases correct?
complex-analysis
complex-analysis
asked Nov 14 at 10:23
JDoeDoe
7711513
asked Nov 14 at 10:23
JDoeDoe
7711513
edited Nov 14 at 10:39
asked Nov 14 at 10:23
JDoeDoe
7711513
asked Nov 14 at 10:23
JDoeDoe
7711513
asked Nov 14 at 10:23
JDoeDoe
7711513
7711513
$arctanfrac{v(t)}{u(t)}$ might not be defined, $u$ might have zeros, the definition of $theta$ is not that lucky, better use the argument function.
– Peter Melech
Nov 14 at 11:16
add a comment |
$arctanfrac{v(t)}{u(t)}$ might not be defined, $u$ might have zeros, the definition of $theta$ is not that lucky, better use the argument function.
– Peter Melech
Nov 14 at 11:16
$arctanfrac{v(t)}{u(t)}$ might not be defined, $u$ might have zeros, the definition of $theta$ is not that lucky, better use the argument function.
– Peter Melech
Nov 14 at 11:16
$arctanfrac{v(t)}{u(t)}$ might not be defined, $u$ might have zeros, the definition of $theta$ is not that lucky, better use the argument function.
– Peter Melech
Nov 14 at 11:16
add a comment |
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$arctanfrac{v(t)}{u(t)}$ might not be defined, $u$ might have zeros, the definition of $theta$ is not that lucky, better use the argument function.
– Peter Melech
Nov 14 at 11:16