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What are the Faber polynomial coefficients?

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The Faber polynomial recurrence relation is given as
$$phi_m+1(z)=frac1c(z phi_m(z)-m c_m-sum_m=0^Mc_m phi_M-m(z) )$$
with the initial values:
$phi_1 = frac1c(z-c_0)$ and $phi_0=1$.

$c_0...c_m$ denote the Laurent expansion coefficients of the mapping from the exterior of the unit disc in the $omega$-plane onto the simply connected exterior of a compact domain $Omega$ in the $z$-plane, $c$ is the transfinite diameter of $Omega$ and $M$ the truncation order to get the polynomial projection.

So say if I were to compute the $5$th order polynomial the recurrence relation would look something like this:
$$phi_5=frac1c(z phi_4-4 c_4-c_0phi_4-c_1phi_3-c_2phi_2-c_3phi_1-c_4phi_0 )$$

Quite obviously if the Laurent expansion coefficients and previous polynomials are known for a certain $z$ the result is a scalar, right ?

I ve heard the term Faber polynomial coefficient used like here, here and Discroll talks about computing it with the SC-toolbox, but I do not understand what it actually is.
What exactly does it mean ? The Faber coefficient is a time valued function gained from computing
$$a_m = frac12piintfracg(Psi (omega))omega^m+1domega$$
with $Psi$ denoting the map I mentioned earlier.
What's the difference between that and the polynomial coefficient?

complex-analysis conformal-geometry

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asked Mar 13 at 10:47

Tony_VTony_V

62

New contributor

Tony_V is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

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0

1

$begingroup$

The Faber polynomial recurrence relation is given as
$$phi_m+1(z)=frac1c(z phi_m(z)-m c_m-sum_m=0^Mc_m phi_M-m(z) )$$
with the initial values:
$phi_1 = frac1c(z-c_0)$ and $phi_0=1$.

$c_0...c_m$ denote the Laurent expansion coefficients of the mapping from the exterior of the unit disc in the $omega$-plane onto the simply connected exterior of a compact domain $Omega$ in the $z$-plane, $c$ is the transfinite diameter of $Omega$ and $M$ the truncation order to get the polynomial projection.

So say if I were to compute the $5$th order polynomial the recurrence relation would look something like this:
$$phi_5=frac1c(z phi_4-4 c_4-c_0phi_4-c_1phi_3-c_2phi_2-c_3phi_1-c_4phi_0 )$$

Quite obviously if the Laurent expansion coefficients and previous polynomials are known for a certain $z$ the result is a scalar, right ?

I ve heard the term Faber polynomial coefficient used like here, here and Discroll talks about computing it with the SC-toolbox, but I do not understand what it actually is.
What exactly does it mean ? The Faber coefficient is a time valued function gained from computing
$$a_m = frac12piintfracg(Psi (omega))omega^m+1domega$$
with $Psi$ denoting the map I mentioned earlier.
What's the difference between that and the polynomial coefficient?

complex-analysis conformal-geometry

share|cite|improve this question

asked Mar 13 at 10:47

Tony_VTony_V

62

New contributor

Tony_V is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

The Faber polynomial recurrence relation is given as
$$phi_m+1(z)=frac1c(z phi_m(z)-m c_m-sum_m=0^Mc_m phi_M-m(z) )$$
with the initial values:
$phi_1 = frac1c(z-c_0)$ and $phi_0=1$.

$c_0...c_m$ denote the Laurent expansion coefficients of the mapping from the exterior of the unit disc in the $omega$-plane onto the simply connected exterior of a compact domain $Omega$ in the $z$-plane, $c$ is the transfinite diameter of $Omega$ and $M$ the truncation order to get the polynomial projection.

So say if I were to compute the $5$th order polynomial the recurrence relation would look something like this:
$$phi_5=frac1c(z phi_4-4 c_4-c_0phi_4-c_1phi_3-c_2phi_2-c_3phi_1-c_4phi_0 )$$

Quite obviously if the Laurent expansion coefficients and previous polynomials are known for a certain $z$ the result is a scalar, right ?

I ve heard the term Faber polynomial coefficient used like here, here and Discroll talks about computing it with the SC-toolbox, but I do not understand what it actually is.
What exactly does it mean ? The Faber coefficient is a time valued function gained from computing
$$a_m = frac12piintfracg(Psi (omega))omega^m+1domega$$
with $Psi$ denoting the map I mentioned earlier.
What's the difference between that and the polynomial coefficient?

complex-analysis conformal-geometry

share|cite|improve this question

asked Mar 13 at 10:47

Tony_VTony_V

62

New contributor

Tony_V is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

asked Mar 13 at 10:47

Tony_VTony_V

62

New contributor

Tony_V is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked Mar 13 at 10:47

Tony_VTony_V

62

New contributor

Tony_V is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 13 at 10:47

Tony_VTony_V

62

New contributor

Tony_V is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 13 at 10:47

Tony_VTony_V

62