Ahlfors page 171 Poisson Integral
Tl;dr : compute the last integral with $z$ fixed.
If $C_1$ and $C_2$ are complementary arcs on the unit circlw, set $U = 1$ on $C_1$ and $U=0$ on $C_2$. Let $0 leq theta_0 < theta_1 leq 2 pi$ be the endpoints of $C_1$. Find $P_U(z) = frac{1-|z|^2}{2 pi} int_{theta_0}^{theta_1} frac 1{|e^{itheta}-z|^2} d theta$ explicitly.
I am not sure how to do this here. Clearly we cannot just use partial fraction decomposition because real-integrating a complex function wouldn't make sense, nor can we use residue theorem since it's just an arc.
Clearly we only have to find $$int_{theta_0}^{theta_1} frac1{|e^{itheta}-z|^2} dtheta$$ with a constant $z=a+ib$ inside the unit disk. If $|z| = 0$ or $1$ noth are easy to solve, the ltter being consequence of Poisson kernel approximating boundary values. Hence assume otherwise. haved tried writing out $e^{itheta} = cos theta + i sin theta$ but it got way too complicated. Is there a nice way t do this integral?
integration complex-analysis complex-integration riemann-integration poissons-equation
asked Nov 25 at 8:37
Cute Brownie
995416
add a comment |
Tl;dr : compute the last integral with $z$ fixed.
If $C_1$ and $C_2$ are complementary arcs on the unit circlw, set $U = 1$ on $C_1$ and $U=0$ on $C_2$. Let $0 leq theta_0 < theta_1 leq 2 pi$ be the endpoints of $C_1$. Find $P_U(z) = frac{1-|z|^2}{2 pi} int_{theta_0}^{theta_1} frac 1{|e^{itheta}-z|^2} d theta$ explicitly.
I am not sure how to do this here. Clearly we cannot just use partial fraction decomposition because real-integrating a complex function wouldn't make sense, nor can we use residue theorem since it's just an arc.
Clearly we only have to find $$int_{theta_0}^{theta_1} frac1{|e^{itheta}-z|^2} dtheta$$ with a constant $z=a+ib$ inside the unit disk. If $|z| = 0$ or $1$ noth are easy to solve, the ltter being consequence of Poisson kernel approximating boundary values. Hence assume otherwise. haved tried writing out $e^{itheta} = cos theta + i sin theta$ but it got way too complicated. Is there a nice way t do this integral?
integration complex-analysis complex-integration riemann-integration poissons-equation
asked Nov 25 at 8:37
Cute Brownie
995416
The integrand is real. Integrate it real-ly.
– Szeto
Nov 25 at 8:53
add a comment |
Tl;dr : compute the last integral with $z$ fixed.
If $C_1$ and $C_2$ are complementary arcs on the unit circlw, set $U = 1$ on $C_1$ and $U=0$ on $C_2$. Let $0 leq theta_0 < theta_1 leq 2 pi$ be the endpoints of $C_1$. Find $P_U(z) = frac{1-|z|^2}{2 pi} int_{theta_0}^{theta_1} frac 1{|e^{itheta}-z|^2} d theta$ explicitly.
I am not sure how to do this here. Clearly we cannot just use partial fraction decomposition because real-integrating a complex function wouldn't make sense, nor can we use residue theorem since it's just an arc.
Clearly we only have to find $$int_{theta_0}^{theta_1} frac1{|e^{itheta}-z|^2} dtheta$$ with a constant $z=a+ib$ inside the unit disk. If $|z| = 0$ or $1$ noth are easy to solve, the ltter being consequence of Poisson kernel approximating boundary values. Hence assume otherwise. haved tried writing out $e^{itheta} = cos theta + i sin theta$ but it got way too complicated. Is there a nice way t do this integral?
integration complex-analysis complex-integration riemann-integration poissons-equation
asked Nov 25 at 8:37
Cute Brownie
995416
Tl;dr : compute the last integral with $z$ fixed.
If $C_1$ and $C_2$ are complementary arcs on the unit circlw, set $U = 1$ on $C_1$ and $U=0$ on $C_2$. Let $0 leq theta_0 < theta_1 leq 2 pi$ be the endpoints of $C_1$. Find $P_U(z) = frac{1-|z|^2}{2 pi} int_{theta_0}^{theta_1} frac 1{|e^{itheta}-z|^2} d theta$ explicitly.
I am not sure how to do this here. Clearly we cannot just use partial fraction decomposition because real-integrating a complex function wouldn't make sense, nor can we use residue theorem since it's just an arc.
Clearly we only have to find $$int_{theta_0}^{theta_1} frac1{|e^{itheta}-z|^2} dtheta$$ with a constant $z=a+ib$ inside the unit disk. If $|z| = 0$ or $1$ noth are easy to solve, the ltter being consequence of Poisson kernel approximating boundary values. Hence assume otherwise. haved tried writing out $e^{itheta} = cos theta + i sin theta$ but it got way too complicated. Is there a nice way t do this integral?
integration complex-analysis complex-integration riemann-integration poissons-equation
integration complex-analysis complex-integration riemann-integration poissons-equation
asked Nov 25 at 8:37
Cute Brownie
995416
asked Nov 25 at 8:37
Cute Brownie
995416
edited Nov 25 at 8:43
asked Nov 25 at 8:37
Cute Brownie
995416
asked Nov 25 at 8:37
Cute Brownie
995416
asked Nov 25 at 8:37
Cute Brownie
995416
995416
The integrand is real. Integrate it real-ly.
– Szeto
Nov 25 at 8:53
add a comment |
The integrand is real. Integrate it real-ly.
– Szeto
Nov 25 at 8:53
The integrand is real. Integrate it real-ly.
– Szeto
Nov 25 at 8:53
The integrand is real. Integrate it real-ly.
– Szeto
Nov 25 at 8:53
add a comment |
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The integrand is real. Integrate it real-ly.
– Szeto
Nov 25 at 8:53