Plotting odd complex functions w/o computer help
I am evaluating a complex integral that utilises the Cauchy Integral Formula and its properties.
In the book I'm reading, they give examples of evaluating integrals using CIT by graphing them, which really does help to see if points of a function is analytic in a certain domain.
For example, in evaluating this integral - how would I be able to plot the function shown in this integral, where C is a circle $|z|=1$ traversed once counter clockwise. $$int_{C}frac{z+i}{z^3+2z^2}$$
Thanks.
complex-analysis complex-numbers
asked 4 hours ago
Dr.Doofus
9210
add a comment |
I am evaluating a complex integral that utilises the Cauchy Integral Formula and its properties.
In the book I'm reading, they give examples of evaluating integrals using CIT by graphing them, which really does help to see if points of a function is analytic in a certain domain.
For example, in evaluating this integral - how would I be able to plot the function shown in this integral, where C is a circle $|z|=1$ traversed once counter clockwise. $$int_{C}frac{z+i}{z^3+2z^2}$$
Thanks.
complex-analysis complex-numbers
asked 4 hours ago
Dr.Doofus
9210
You may be able to find ways of visualizing complex mappings but in general there is no way to plot the graph of a function from $mathbb{C}to mathbb{C}$ analogously to graphs of functions from $mathbb{R} to mathbb{R}$ (or up to 3-space). The point $(z, f(z))$ is in some sense a point of $mathbb{R}^4$ which our brains find difficult to visualize...
– LoveTooNap29
4 hours ago
I understand. This is something that I will come to terms with. I'm used to plotting basic real functions. Thanks for that. So in a case like this, how would one go about solving it? I understand that the integrand fails to be analytic point z = 0, but where to from there?
– Dr.Doofus
4 hours ago
Indeed, it would be relatively straightforward to apply residue theorem (although probably tedious) but it appears your only given tool is Cauchy’s integral formula, correct? Just want to make sure before I offer a sketch of a solution.
– LoveTooNap29
4 hours ago
Yes, sir. It's how they want us to go about it.
– Dr.Doofus
4 hours ago
add a comment |
I am evaluating a complex integral that utilises the Cauchy Integral Formula and its properties.
In the book I'm reading, they give examples of evaluating integrals using CIT by graphing them, which really does help to see if points of a function is analytic in a certain domain.
For example, in evaluating this integral - how would I be able to plot the function shown in this integral, where C is a circle $|z|=1$ traversed once counter clockwise. $$int_{C}frac{z+i}{z^3+2z^2}$$
Thanks.
complex-analysis complex-numbers
asked 4 hours ago
Dr.Doofus
9210
I am evaluating a complex integral that utilises the Cauchy Integral Formula and its properties.
In the book I'm reading, they give examples of evaluating integrals using CIT by graphing them, which really does help to see if points of a function is analytic in a certain domain.
For example, in evaluating this integral - how would I be able to plot the function shown in this integral, where C is a circle $|z|=1$ traversed once counter clockwise. $$int_{C}frac{z+i}{z^3+2z^2}$$
Thanks.
complex-analysis complex-numbers
complex-analysis complex-numbers
asked 4 hours ago
Dr.Doofus
9210
asked 4 hours ago
Dr.Doofus
9210
asked 4 hours ago
Dr.Doofus
9210
asked 4 hours ago
Dr.Doofus
9210
asked 4 hours ago
Dr.Doofus
9210
9210
You may be able to find ways of visualizing complex mappings but in general there is no way to plot the graph of a function from $mathbb{C}to mathbb{C}$ analogously to graphs of functions from $mathbb{R} to mathbb{R}$ (or up to 3-space). The point $(z, f(z))$ is in some sense a point of $mathbb{R}^4$ which our brains find difficult to visualize...
– LoveTooNap29
4 hours ago
I understand. This is something that I will come to terms with. I'm used to plotting basic real functions. Thanks for that. So in a case like this, how would one go about solving it? I understand that the integrand fails to be analytic point z = 0, but where to from there?
– Dr.Doofus
4 hours ago
Indeed, it would be relatively straightforward to apply residue theorem (although probably tedious) but it appears your only given tool is Cauchy’s integral formula, correct? Just want to make sure before I offer a sketch of a solution.
– LoveTooNap29
4 hours ago
Yes, sir. It's how they want us to go about it.
– Dr.Doofus
4 hours ago
add a comment |
You may be able to find ways of visualizing complex mappings but in general there is no way to plot the graph of a function from $mathbb{C}to mathbb{C}$ analogously to graphs of functions from $mathbb{R} to mathbb{R}$ (or up to 3-space). The point $(z, f(z))$ is in some sense a point of $mathbb{R}^4$ which our brains find difficult to visualize...
– LoveTooNap29
4 hours ago
I understand. This is something that I will come to terms with. I'm used to plotting basic real functions. Thanks for that. So in a case like this, how would one go about solving it? I understand that the integrand fails to be analytic point z = 0, but where to from there?
– Dr.Doofus
4 hours ago
Indeed, it would be relatively straightforward to apply residue theorem (although probably tedious) but it appears your only given tool is Cauchy’s integral formula, correct? Just want to make sure before I offer a sketch of a solution.
– LoveTooNap29
4 hours ago
Yes, sir. It's how they want us to go about it.
– Dr.Doofus
4 hours ago
You may be able to find ways of visualizing complex mappings but in general there is no way to plot the graph of a function from $mathbb{C}to mathbb{C}$ analogously to graphs of functions from $mathbb{R} to mathbb{R}$ (or up to 3-space). The point $(z, f(z))$ is in some sense a point of $mathbb{R}^4$ which our brains find difficult to visualize...
– LoveTooNap29
4 hours ago
You may be able to find ways of visualizing complex mappings but in general there is no way to plot the graph of a function from $mathbb{C}to mathbb{C}$ analogously to graphs of functions from $mathbb{R} to mathbb{R}$ (or up to 3-space). The point $(z, f(z))$ is in some sense a point of $mathbb{R}^4$ which our brains find difficult to visualize...
– LoveTooNap29
4 hours ago
I understand. This is something that I will come to terms with. I'm used to plotting basic real functions. Thanks for that. So in a case like this, how would one go about solving it? I understand that the integrand fails to be analytic point z = 0, but where to from there?
– Dr.Doofus
4 hours ago
I understand. This is something that I will come to terms with. I'm used to plotting basic real functions. Thanks for that. So in a case like this, how would one go about solving it? I understand that the integrand fails to be analytic point z = 0, but where to from there?
– Dr.Doofus
4 hours ago
Indeed, it would be relatively straightforward to apply residue theorem (although probably tedious) but it appears your only given tool is Cauchy’s integral formula, correct? Just want to make sure before I offer a sketch of a solution.
– LoveTooNap29
4 hours ago
Indeed, it would be relatively straightforward to apply residue theorem (although probably tedious) but it appears your only given tool is Cauchy’s integral formula, correct? Just want to make sure before I offer a sketch of a solution.
– LoveTooNap29
4 hours ago
Yes, sir. It's how they want us to go about it.
– Dr.Doofus
4 hours ago
Yes, sir. It's how they want us to go about it.
– Dr.Doofus
4 hours ago
add a comment |
2 Answers
2
active
oldest
votes
The idea is that you only need to know where the function is singular
In your case it is at $z= 0$ and $z = -2$, but the second point is outside the region of integration, so we do not care about it
$$
oint_C frac{z + i}{z^2(z + 2)} {rm d}z= oint_C frac{f(z)}{z^2}{rm d}z
$$
where the function
$$
f(z) = frac{z + i}{z + 2}
$$
is well behaved in the region $|z| leq 1$. The problem is at $z = 0$, where you have second order singularity, so you can use Cauchy's integral form
$$
left.frac{{rm d}^n f}{{rm d}z^n}right|_{z = 0} = frac{n!}{2pi i}oint_C frac{f(z)}{z^{n + 1}}{rm d}z
$$
with $n = 1$ which yields
$$
oint_Cfrac{z + i}{z^2(z + 2)} = 2 pi i frac{{rm d}}{{rm d}z}left(frac{z + i}{z + 2}right)_{z = 0} = 2pi i left(frac{1}{2} - frac{i}{4} right)
$$
answered 3 hours ago
caverac
13.9k21130
This is great. Thank you. Better than the provided solutions!
– Dr.Doofus
3 hours ago
Ah it looks like you beat me to it! Excellent visual too. Good to at least confirm by someone else's work that I can still recall this material correctly.
– LoveTooNap29
3 hours ago
add a comment |
Sketch:
Note that the pole at $z=0$ has order $2$. Use the slightly generalized Cauchy formula for derivatives,
$$f^{(n)}(a)=frac{n!}{2pi i}oint frac{f(z)}{(z-a)^{n+1}} mathrm{d}z,$$
Write the original integrand as $g(z)$ and identify $f(z)$ from $g(z)=f(z)/z^2$ in your case for $n=1$ and $a=0$. Then compute the first derivative of $f$ at $a=0$ to compute the original integral (remember to multiply this by $2pi i$). This yields $pi (frac12 +i)$.
Be sure to check all hypotheses of the Cauchy integral formula though to justify this rigorously, though (that $f$ is holomorphic inside the curve is required, not $g$, etc). Comment if you want some steps expanded, but preferably with where you got stuck also, or if you spot any typos, etc. It has been some time since I've gotten my hands dirty with contour integrals with only Cauchy's integral formula, however, the example on wikipedia is rather illustrative, though the poles there are only of order 1, unlike here.
answered 3 hours ago
LoveTooNap29
1,0241613
Thank you. I genuinely appreciate it.
– Dr.Doofus
3 hours ago
Sorry, didn't know you were working on an answer as well (+1)
– caverac
3 hours ago
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
The idea is that you only need to know where the function is singular
In your case it is at $z= 0$ and $z = -2$, but the second point is outside the region of integration, so we do not care about it
$$
oint_C frac{z + i}{z^2(z + 2)} {rm d}z= oint_C frac{f(z)}{z^2}{rm d}z
$$
where the function
$$
f(z) = frac{z + i}{z + 2}
$$
is well behaved in the region $|z| leq 1$. The problem is at $z = 0$, where you have second order singularity, so you can use Cauchy's integral form
$$
left.frac{{rm d}^n f}{{rm d}z^n}right|_{z = 0} = frac{n!}{2pi i}oint_C frac{f(z)}{z^{n + 1}}{rm d}z
$$
with $n = 1$ which yields
$$
oint_Cfrac{z + i}{z^2(z + 2)} = 2 pi i frac{{rm d}}{{rm d}z}left(frac{z + i}{z + 2}right)_{z = 0} = 2pi i left(frac{1}{2} - frac{i}{4} right)
$$
answered 3 hours ago
caverac
13.9k21130
This is great. Thank you. Better than the provided solutions!
– Dr.Doofus
3 hours ago
Ah it looks like you beat me to it! Excellent visual too. Good to at least confirm by someone else's work that I can still recall this material correctly.
– LoveTooNap29
3 hours ago
add a comment |
The idea is that you only need to know where the function is singular
In your case it is at $z= 0$ and $z = -2$, but the second point is outside the region of integration, so we do not care about it
$$
oint_C frac{z + i}{z^2(z + 2)} {rm d}z= oint_C frac{f(z)}{z^2}{rm d}z
$$
where the function
$$
f(z) = frac{z + i}{z + 2}
$$
is well behaved in the region $|z| leq 1$. The problem is at $z = 0$, where you have second order singularity, so you can use Cauchy's integral form
$$
left.frac{{rm d}^n f}{{rm d}z^n}right|_{z = 0} = frac{n!}{2pi i}oint_C frac{f(z)}{z^{n + 1}}{rm d}z
$$
with $n = 1$ which yields
$$
oint_Cfrac{z + i}{z^2(z + 2)} = 2 pi i frac{{rm d}}{{rm d}z}left(frac{z + i}{z + 2}right)_{z = 0} = 2pi i left(frac{1}{2} - frac{i}{4} right)
$$
answered 3 hours ago
caverac
13.9k21130
This is great. Thank you. Better than the provided solutions!
– Dr.Doofus
3 hours ago
Ah it looks like you beat me to it! Excellent visual too. Good to at least confirm by someone else's work that I can still recall this material correctly.
– LoveTooNap29
3 hours ago
add a comment |
The idea is that you only need to know where the function is singular
In your case it is at $z= 0$ and $z = -2$, but the second point is outside the region of integration, so we do not care about it
$$
oint_C frac{z + i}{z^2(z + 2)} {rm d}z= oint_C frac{f(z)}{z^2}{rm d}z
$$
where the function
$$
f(z) = frac{z + i}{z + 2}
$$
is well behaved in the region $|z| leq 1$. The problem is at $z = 0$, where you have second order singularity, so you can use Cauchy's integral form
$$
left.frac{{rm d}^n f}{{rm d}z^n}right|_{z = 0} = frac{n!}{2pi i}oint_C frac{f(z)}{z^{n + 1}}{rm d}z
$$
with $n = 1$ which yields
$$
oint_Cfrac{z + i}{z^2(z + 2)} = 2 pi i frac{{rm d}}{{rm d}z}left(frac{z + i}{z + 2}right)_{z = 0} = 2pi i left(frac{1}{2} - frac{i}{4} right)
$$
answered 3 hours ago
caverac
13.9k21130
The idea is that you only need to know where the function is singular
In your case it is at $z= 0$ and $z = -2$, but the second point is outside the region of integration, so we do not care about it
$$
oint_C frac{z + i}{z^2(z + 2)} {rm d}z= oint_C frac{f(z)}{z^2}{rm d}z
$$
where the function
$$
f(z) = frac{z + i}{z + 2}
$$
is well behaved in the region $|z| leq 1$. The problem is at $z = 0$, where you have second order singularity, so you can use Cauchy's integral form
$$
left.frac{{rm d}^n f}{{rm d}z^n}right|_{z = 0} = frac{n!}{2pi i}oint_C frac{f(z)}{z^{n + 1}}{rm d}z
$$
with $n = 1$ which yields
$$
oint_Cfrac{z + i}{z^2(z + 2)} = 2 pi i frac{{rm d}}{{rm d}z}left(frac{z + i}{z + 2}right)_{z = 0} = 2pi i left(frac{1}{2} - frac{i}{4} right)
$$
answered 3 hours ago
caverac
13.9k21130
answered 3 hours ago
caverac
13.9k21130
answered 3 hours ago
caverac
13.9k21130
answered 3 hours ago
caverac
13.9k21130
13.9k21130
This is great. Thank you. Better than the provided solutions!
– Dr.Doofus
3 hours ago
Ah it looks like you beat me to it! Excellent visual too. Good to at least confirm by someone else's work that I can still recall this material correctly.
– LoveTooNap29
3 hours ago
add a comment |
This is great. Thank you. Better than the provided solutions!
– Dr.Doofus
3 hours ago
Ah it looks like you beat me to it! Excellent visual too. Good to at least confirm by someone else's work that I can still recall this material correctly.
– LoveTooNap29
3 hours ago
This is great. Thank you. Better than the provided solutions!
– Dr.Doofus
3 hours ago
This is great. Thank you. Better than the provided solutions!
– Dr.Doofus
3 hours ago
Ah it looks like you beat me to it! Excellent visual too. Good to at least confirm by someone else's work that I can still recall this material correctly.
– LoveTooNap29
3 hours ago
Ah it looks like you beat me to it! Excellent visual too. Good to at least confirm by someone else's work that I can still recall this material correctly.
– LoveTooNap29
3 hours ago
add a comment |
Sketch:
Note that the pole at $z=0$ has order $2$. Use the slightly generalized Cauchy formula for derivatives,
$$f^{(n)}(a)=frac{n!}{2pi i}oint frac{f(z)}{(z-a)^{n+1}} mathrm{d}z,$$
Write the original integrand as $g(z)$ and identify $f(z)$ from $g(z)=f(z)/z^2$ in your case for $n=1$ and $a=0$. Then compute the first derivative of $f$ at $a=0$ to compute the original integral (remember to multiply this by $2pi i$). This yields $pi (frac12 +i)$.
Be sure to check all hypotheses of the Cauchy integral formula though to justify this rigorously, though (that $f$ is holomorphic inside the curve is required, not $g$, etc). Comment if you want some steps expanded, but preferably with where you got stuck also, or if you spot any typos, etc. It has been some time since I've gotten my hands dirty with contour integrals with only Cauchy's integral formula, however, the example on wikipedia is rather illustrative, though the poles there are only of order 1, unlike here.
answered 3 hours ago
LoveTooNap29
1,0241613
Thank you. I genuinely appreciate it.
– Dr.Doofus
3 hours ago
Sorry, didn't know you were working on an answer as well (+1)
– caverac
3 hours ago
add a comment |
Sketch:
Note that the pole at $z=0$ has order $2$. Use the slightly generalized Cauchy formula for derivatives,
$$f^{(n)}(a)=frac{n!}{2pi i}oint frac{f(z)}{(z-a)^{n+1}} mathrm{d}z,$$
Write the original integrand as $g(z)$ and identify $f(z)$ from $g(z)=f(z)/z^2$ in your case for $n=1$ and $a=0$. Then compute the first derivative of $f$ at $a=0$ to compute the original integral (remember to multiply this by $2pi i$). This yields $pi (frac12 +i)$.
Be sure to check all hypotheses of the Cauchy integral formula though to justify this rigorously, though (that $f$ is holomorphic inside the curve is required, not $g$, etc). Comment if you want some steps expanded, but preferably with where you got stuck also, or if you spot any typos, etc. It has been some time since I've gotten my hands dirty with contour integrals with only Cauchy's integral formula, however, the example on wikipedia is rather illustrative, though the poles there are only of order 1, unlike here.
answered 3 hours ago
LoveTooNap29
1,0241613
Thank you. I genuinely appreciate it.
– Dr.Doofus
3 hours ago
Sorry, didn't know you were working on an answer as well (+1)
– caverac
3 hours ago
add a comment |
Sketch:
Note that the pole at $z=0$ has order $2$. Use the slightly generalized Cauchy formula for derivatives,
$$f^{(n)}(a)=frac{n!}{2pi i}oint frac{f(z)}{(z-a)^{n+1}} mathrm{d}z,$$
Write the original integrand as $g(z)$ and identify $f(z)$ from $g(z)=f(z)/z^2$ in your case for $n=1$ and $a=0$. Then compute the first derivative of $f$ at $a=0$ to compute the original integral (remember to multiply this by $2pi i$). This yields $pi (frac12 +i)$.
Be sure to check all hypotheses of the Cauchy integral formula though to justify this rigorously, though (that $f$ is holomorphic inside the curve is required, not $g$, etc). Comment if you want some steps expanded, but preferably with where you got stuck also, or if you spot any typos, etc. It has been some time since I've gotten my hands dirty with contour integrals with only Cauchy's integral formula, however, the example on wikipedia is rather illustrative, though the poles there are only of order 1, unlike here.
answered 3 hours ago
LoveTooNap29
1,0241613
Sketch:
Note that the pole at $z=0$ has order $2$. Use the slightly generalized Cauchy formula for derivatives,
$$f^{(n)}(a)=frac{n!}{2pi i}oint frac{f(z)}{(z-a)^{n+1}} mathrm{d}z,$$
Write the original integrand as $g(z)$ and identify $f(z)$ from $g(z)=f(z)/z^2$ in your case for $n=1$ and $a=0$. Then compute the first derivative of $f$ at $a=0$ to compute the original integral (remember to multiply this by $2pi i$). This yields $pi (frac12 +i)$.
Be sure to check all hypotheses of the Cauchy integral formula though to justify this rigorously, though (that $f$ is holomorphic inside the curve is required, not $g$, etc). Comment if you want some steps expanded, but preferably with where you got stuck also, or if you spot any typos, etc. It has been some time since I've gotten my hands dirty with contour integrals with only Cauchy's integral formula, however, the example on wikipedia is rather illustrative, though the poles there are only of order 1, unlike here.
answered 3 hours ago
LoveTooNap29
1,0241613
answered 3 hours ago
LoveTooNap29
1,0241613
answered 3 hours ago
LoveTooNap29
1,0241613
answered 3 hours ago
LoveTooNap29
1,0241613
1,0241613
Thank you. I genuinely appreciate it.
– Dr.Doofus
3 hours ago
Sorry, didn't know you were working on an answer as well (+1)
– caverac
3 hours ago
add a comment |
Thank you. I genuinely appreciate it.
– Dr.Doofus
3 hours ago
Sorry, didn't know you were working on an answer as well (+1)
– caverac
3 hours ago
Thank you. I genuinely appreciate it.
– Dr.Doofus
3 hours ago
Thank you. I genuinely appreciate it.
– Dr.Doofus
3 hours ago
Sorry, didn't know you were working on an answer as well (+1)
– caverac
3 hours ago
Sorry, didn't know you were working on an answer as well (+1)
– caverac
3 hours ago
add a comment |
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You may be able to find ways of visualizing complex mappings but in general there is no way to plot the graph of a function from $mathbb{C}to mathbb{C}$ analogously to graphs of functions from $mathbb{R} to mathbb{R}$ (or up to 3-space). The point $(z, f(z))$ is in some sense a point of $mathbb{R}^4$ which our brains find difficult to visualize...
– LoveTooNap29
4 hours ago
I understand. This is something that I will come to terms with. I'm used to plotting basic real functions. Thanks for that. So in a case like this, how would one go about solving it? I understand that the integrand fails to be analytic point z = 0, but where to from there?
– Dr.Doofus
4 hours ago
Indeed, it would be relatively straightforward to apply residue theorem (although probably tedious) but it appears your only given tool is Cauchy’s integral formula, correct? Just want to make sure before I offer a sketch of a solution.
– LoveTooNap29
4 hours ago
Yes, sir. It's how they want us to go about it.
– Dr.Doofus
4 hours ago