Mixing
What does $dotplus$ mean when used in the context of contours?
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I am reading David C. Ullrich's book and it says:
If $gamma_1, ..., gamma_n$ are smooth curves and we write
$$Gamma = gamma_1 dotplus cdots dotplus gamma_n$$
that means "$Gamma$ is something over which we can integrate a function, and by definition
$$int_{Gamma} f(z) dz = int_{gamma_1}f(z) dz + cdots + int_{gamma_2}f(z) dz$$
for every $f$ which is continuous on the union of the $gamma_j^*$".
Does $gamma_1 dotplus gamma_2$ just mean to "put the paths together"? Do the paths have to meet up "tip to tail", or any other restrictions?
complex-analysis analysis
asked Jan 30 at 20:11
OviOvi
12.5k1040115
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add a comment |
$begingroup$
I am reading David C. Ullrich's book and it says:
If $gamma_1, ..., gamma_n$ are smooth curves and we write
$$Gamma = gamma_1 dotplus cdots dotplus gamma_n$$
that means "$Gamma$ is something over which we can integrate a function, and by definition
$$int_{Gamma} f(z) dz = int_{gamma_1}f(z) dz + cdots + int_{gamma_2}f(z) dz$$
for every $f$ which is continuous on the union of the $gamma_j^*$".
Does $gamma_1 dotplus gamma_2$ just mean to "put the paths together"? Do the paths have to meet up "tip to tail", or any other restrictions?
complex-analysis analysis
asked Jan 30 at 20:11
OviOvi
12.5k1040115
$endgroup$
$begingroup$
It may mean "put the paths together". The paths don't have to match up in any way. More formally it's an element of the free Abelian groups generated by paths, so basically a set of "$1$-chains" (a la singular homology in algebraic topology).
$endgroup$
– Lord Shark the Unknown
Jan 30 at 20:13
add a comment |
$begingroup$
I am reading David C. Ullrich's book and it says:
If $gamma_1, ..., gamma_n$ are smooth curves and we write
$$Gamma = gamma_1 dotplus cdots dotplus gamma_n$$
that means "$Gamma$ is something over which we can integrate a function, and by definition
$$int_{Gamma} f(z) dz = int_{gamma_1}f(z) dz + cdots + int_{gamma_2}f(z) dz$$
for every $f$ which is continuous on the union of the $gamma_j^*$".
Does $gamma_1 dotplus gamma_2$ just mean to "put the paths together"? Do the paths have to meet up "tip to tail", or any other restrictions?
complex-analysis analysis
asked Jan 30 at 20:11
OviOvi
12.5k1040115
$endgroup$
I am reading David C. Ullrich's book and it says:
If $gamma_1, ..., gamma_n$ are smooth curves and we write
$$Gamma = gamma_1 dotplus cdots dotplus gamma_n$$
that means "$Gamma$ is something over which we can integrate a function, and by definition
$$int_{Gamma} f(z) dz = int_{gamma_1}f(z) dz + cdots + int_{gamma_2}f(z) dz$$
for every $f$ which is continuous on the union of the $gamma_j^*$".
Does $gamma_1 dotplus gamma_2$ just mean to "put the paths together"? Do the paths have to meet up "tip to tail", or any other restrictions?
complex-analysis analysis
complex-analysis analysis
asked Jan 30 at 20:11
OviOvi
12.5k1040115
asked Jan 30 at 20:11
OviOvi
12.5k1040115
asked Jan 30 at 20:11
OviOvi
12.5k1040115
asked Jan 30 at 20:11
OviOvi
12.5k1040115
asked Jan 30 at 20:11
OviOvi
12.5k1040115
12.5k1040115
$begingroup$
It may mean "put the paths together". The paths don't have to match up in any way. More formally it's an element of the free Abelian groups generated by paths, so basically a set of "$1$-chains" (a la singular homology in algebraic topology).
$endgroup$
– Lord Shark the Unknown
Jan 30 at 20:13
add a comment |
$begingroup$
It may mean "put the paths together". The paths don't have to match up in any way. More formally it's an element of the free Abelian groups generated by paths, so basically a set of "$1$-chains" (a la singular homology in algebraic topology).
$endgroup$
– Lord Shark the Unknown
Jan 30 at 20:13
$begingroup$
It may mean "put the paths together". The paths don't have to match up in any way. More formally it's an element of the free Abelian groups generated by paths, so basically a set of "$1$-chains" (a la singular homology in algebraic topology).
$endgroup$
– Lord Shark the Unknown
Jan 30 at 20:13
$begingroup$
It may mean "put the paths together". The paths don't have to match up in any way. More formally it's an element of the free Abelian groups generated by paths, so basically a set of "$1$-chains" (a la singular homology in algebraic topology).
$endgroup$
– Lord Shark the Unknown
Jan 30 at 20:13
add a comment |
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$begingroup$
It may mean "put the paths together". The paths don't have to match up in any way. More formally it's an element of the free Abelian groups generated by paths, so basically a set of "$1$-chains" (a la singular homology in algebraic topology).
$endgroup$
– Lord Shark the Unknown
Jan 30 at 20:13