Contour integral over a non-rectifiable contour
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In chapter 16 of Mathematical Analysis Apostol defines contour integrals in terms of Riemann-Stieltjes integrals, specifically:
$$int_gamma f = int_a^b f[gamma(t)],dgamma(t)$$
whenever the Riemann-Stieltjes integral on the right exists. My question would be "Find necessary and sufficient conditions on a contour $gamma$ to ensure that the Riemann-Stieltjes integral defined above (and thus the contour integral) exists for all continuous complex functions $f$."
It is well-known that requiring $gamma$ to be rectifiable is a sufficient condition for the contour integral to exist. But is it necessary? Is it the case that for every non-rectifiable curve $gamma$ we can find a continuous function $f$ for which $int_gamma f$ (as defined above) does not exist? If it is not a necessary condition, what is?
complex-analysis contour-integration
asked Dec 14 '18 at 20:23
A.C.A.C.
112
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add a comment |
$begingroup$
In chapter 16 of Mathematical Analysis Apostol defines contour integrals in terms of Riemann-Stieltjes integrals, specifically:
$$int_gamma f = int_a^b f[gamma(t)],dgamma(t)$$
whenever the Riemann-Stieltjes integral on the right exists. My question would be "Find necessary and sufficient conditions on a contour $gamma$ to ensure that the Riemann-Stieltjes integral defined above (and thus the contour integral) exists for all continuous complex functions $f$."
It is well-known that requiring $gamma$ to be rectifiable is a sufficient condition for the contour integral to exist. But is it necessary? Is it the case that for every non-rectifiable curve $gamma$ we can find a continuous function $f$ for which $int_gamma f$ (as defined above) does not exist? If it is not a necessary condition, what is?
complex-analysis contour-integration
asked Dec 14 '18 at 20:23
A.C.A.C.
112
$endgroup$
$begingroup$
If $f(x) = 1$ then for every tagged partition of the interval [a,b] the finite Riemann sum has the form $sum_{i=1}^n(gamma(x_i) - gamma(x_{i-1}))$ which is a telescoping sum that collapses to $gamma(x_n) - gamma(x_0) = gamma(b) - gamma(a)$. Non-rectifiability only means that partitions can be found that make $sum_{i=1}^n|gamma(x_i) - gamma(x_{i-1})|$ arbitrarily high.
$endgroup$
– A.C.
Dec 14 '18 at 21:05
add a comment |
$begingroup$
In chapter 16 of Mathematical Analysis Apostol defines contour integrals in terms of Riemann-Stieltjes integrals, specifically:
$$int_gamma f = int_a^b f[gamma(t)],dgamma(t)$$
whenever the Riemann-Stieltjes integral on the right exists. My question would be "Find necessary and sufficient conditions on a contour $gamma$ to ensure that the Riemann-Stieltjes integral defined above (and thus the contour integral) exists for all continuous complex functions $f$."
It is well-known that requiring $gamma$ to be rectifiable is a sufficient condition for the contour integral to exist. But is it necessary? Is it the case that for every non-rectifiable curve $gamma$ we can find a continuous function $f$ for which $int_gamma f$ (as defined above) does not exist? If it is not a necessary condition, what is?
complex-analysis contour-integration
asked Dec 14 '18 at 20:23
A.C.A.C.
112
$endgroup$
In chapter 16 of Mathematical Analysis Apostol defines contour integrals in terms of Riemann-Stieltjes integrals, specifically:
$$int_gamma f = int_a^b f[gamma(t)],dgamma(t)$$
whenever the Riemann-Stieltjes integral on the right exists. My question would be "Find necessary and sufficient conditions on a contour $gamma$ to ensure that the Riemann-Stieltjes integral defined above (and thus the contour integral) exists for all continuous complex functions $f$."
It is well-known that requiring $gamma$ to be rectifiable is a sufficient condition for the contour integral to exist. But is it necessary? Is it the case that for every non-rectifiable curve $gamma$ we can find a continuous function $f$ for which $int_gamma f$ (as defined above) does not exist? If it is not a necessary condition, what is?
complex-analysis contour-integration
complex-analysis contour-integration
asked Dec 14 '18 at 20:23
A.C.A.C.
112
asked Dec 14 '18 at 20:23
A.C.A.C.
112
asked Dec 14 '18 at 20:23
A.C.A.C.
112
asked Dec 14 '18 at 20:23
A.C.A.C.
112
asked Dec 14 '18 at 20:23
A.C.A.C.
112
112
$begingroup$
If $f(x) = 1$ then for every tagged partition of the interval [a,b] the finite Riemann sum has the form $sum_{i=1}^n(gamma(x_i) - gamma(x_{i-1}))$ which is a telescoping sum that collapses to $gamma(x_n) - gamma(x_0) = gamma(b) - gamma(a)$. Non-rectifiability only means that partitions can be found that make $sum_{i=1}^n|gamma(x_i) - gamma(x_{i-1})|$ arbitrarily high.
$endgroup$
– A.C.
Dec 14 '18 at 21:05
add a comment |
$begingroup$
If $f(x) = 1$ then for every tagged partition of the interval [a,b] the finite Riemann sum has the form $sum_{i=1}^n(gamma(x_i) - gamma(x_{i-1}))$ which is a telescoping sum that collapses to $gamma(x_n) - gamma(x_0) = gamma(b) - gamma(a)$. Non-rectifiability only means that partitions can be found that make $sum_{i=1}^n|gamma(x_i) - gamma(x_{i-1})|$ arbitrarily high.
$endgroup$
– A.C.
Dec 14 '18 at 21:05
$begingroup$
If $f(x) = 1$ then for every tagged partition of the interval [a,b] the finite Riemann sum has the form $sum_{i=1}^n(gamma(x_i) - gamma(x_{i-1}))$ which is a telescoping sum that collapses to $gamma(x_n) - gamma(x_0) = gamma(b) - gamma(a)$. Non-rectifiability only means that partitions can be found that make $sum_{i=1}^n|gamma(x_i) - gamma(x_{i-1})|$ arbitrarily high.
$endgroup$
– A.C.
Dec 14 '18 at 21:05
$begingroup$
If $f(x) = 1$ then for every tagged partition of the interval [a,b] the finite Riemann sum has the form $sum_{i=1}^n(gamma(x_i) - gamma(x_{i-1}))$ which is a telescoping sum that collapses to $gamma(x_n) - gamma(x_0) = gamma(b) - gamma(a)$. Non-rectifiability only means that partitions can be found that make $sum_{i=1}^n|gamma(x_i) - gamma(x_{i-1})|$ arbitrarily high.
$endgroup$
– A.C.
Dec 14 '18 at 21:05
add a comment |
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$begingroup$
If $f(x) = 1$ then for every tagged partition of the interval [a,b] the finite Riemann sum has the form $sum_{i=1}^n(gamma(x_i) - gamma(x_{i-1}))$ which is a telescoping sum that collapses to $gamma(x_n) - gamma(x_0) = gamma(b) - gamma(a)$. Non-rectifiability only means that partitions can be found that make $sum_{i=1}^n|gamma(x_i) - gamma(x_{i-1})|$ arbitrarily high.
$endgroup$
– A.C.
Dec 14 '18 at 21:05