Complex integral $int{z^noverline{z^m}}dz$ over $|z|<1$
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Let $D={zinBbb{C}, |z|<1}$ and $P_n:DrightarrowBbb{C}$ such that $P_n=sqrt{n+1over{pi}}z^n$. I need to show ${P_n: ninBbb{N}}$ is an orthonormal system in $L^2(D)$.
$<P_n,P_m> = int_D{P_noverline{P_m}dz}$ on $L^2(D)$.
So $<P_n,P_m> = 1$ if $n=m$ and $<P_n,P_m> = 0$ if $nnot=m$.
I am having some problems with the calculation of $int{z^noverline{z^m}dz}$ over $|z|<1$.
Could you help me with this problem please?
complex-analysis functional-analysis
asked Nov 16 at 23:02
Vitiello
887
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up vote
0
down vote
favorite
Let $D={zinBbb{C}, |z|<1}$ and $P_n:DrightarrowBbb{C}$ such that $P_n=sqrt{n+1over{pi}}z^n$. I need to show ${P_n: ninBbb{N}}$ is an orthonormal system in $L^2(D)$.
$<P_n,P_m> = int_D{P_noverline{P_m}dz}$ on $L^2(D)$.
So $<P_n,P_m> = 1$ if $n=m$ and $<P_n,P_m> = 0$ if $nnot=m$.
I am having some problems with the calculation of $int{z^noverline{z^m}dz}$ over $|z|<1$.
Could you help me with this problem please?
complex-analysis functional-analysis
asked Nov 16 at 23:02
Vitiello
887
1
Change to polar coordinates.
– user10354138
Nov 16 at 23:08
3
How do you integrate $z^nbar{z}^m$ over the region $D$? I mean this makes no sense: $int_D,z^nbar{z}^m,text{d}z$. Did you want to say $$iint_D,z^nbar{z}^m,text{d}z,text{d}bar{z},?$$
– Batominovski
Nov 16 at 23:18
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $D={zinBbb{C}, |z|<1}$ and $P_n:DrightarrowBbb{C}$ such that $P_n=sqrt{n+1over{pi}}z^n$. I need to show ${P_n: ninBbb{N}}$ is an orthonormal system in $L^2(D)$.
$<P_n,P_m> = int_D{P_noverline{P_m}dz}$ on $L^2(D)$.
So $<P_n,P_m> = 1$ if $n=m$ and $<P_n,P_m> = 0$ if $nnot=m$.
I am having some problems with the calculation of $int{z^noverline{z^m}dz}$ over $|z|<1$.
Could you help me with this problem please?
complex-analysis functional-analysis
asked Nov 16 at 23:02
Vitiello
887
Let $D={zinBbb{C}, |z|<1}$ and $P_n:DrightarrowBbb{C}$ such that $P_n=sqrt{n+1over{pi}}z^n$. I need to show ${P_n: ninBbb{N}}$ is an orthonormal system in $L^2(D)$.
$<P_n,P_m> = int_D{P_noverline{P_m}dz}$ on $L^2(D)$.
So $<P_n,P_m> = 1$ if $n=m$ and $<P_n,P_m> = 0$ if $nnot=m$.
I am having some problems with the calculation of $int{z^noverline{z^m}dz}$ over $|z|<1$.
Could you help me with this problem please?
complex-analysis functional-analysis
complex-analysis functional-analysis
asked Nov 16 at 23:02
Vitiello
887
asked Nov 16 at 23:02
Vitiello
887
asked Nov 16 at 23:02
Vitiello
887
asked Nov 16 at 23:02
Vitiello
887
asked Nov 16 at 23:02
Vitiello
887
887
1
Change to polar coordinates.
– user10354138
Nov 16 at 23:08
3
How do you integrate $z^nbar{z}^m$ over the region $D$? I mean this makes no sense: $int_D,z^nbar{z}^m,text{d}z$. Did you want to say $$iint_D,z^nbar{z}^m,text{d}z,text{d}bar{z},?$$
– Batominovski
Nov 16 at 23:18
add a comment |
1
Change to polar coordinates.
– user10354138
Nov 16 at 23:08
3
How do you integrate $z^nbar{z}^m$ over the region $D$? I mean this makes no sense: $int_D,z^nbar{z}^m,text{d}z$. Did you want to say $$iint_D,z^nbar{z}^m,text{d}z,text{d}bar{z},?$$
– Batominovski
Nov 16 at 23:18
1
1
Change to polar coordinates.
– user10354138
Nov 16 at 23:08
Change to polar coordinates.
– user10354138
Nov 16 at 23:08
3
3
How do you integrate $z^nbar{z}^m$ over the region $D$? I mean this makes no sense: $int_D,z^nbar{z}^m,text{d}z$. Did you want to say $$iint_D,z^nbar{z}^m,text{d}z,text{d}bar{z},?$$
– Batominovski
Nov 16 at 23:18
How do you integrate $z^nbar{z}^m$ over the region $D$? I mean this makes no sense: $int_D,z^nbar{z}^m,text{d}z$. Did you want to say $$iint_D,z^nbar{z}^m,text{d}z,text{d}bar{z},?$$
– Batominovski
Nov 16 at 23:18
add a comment |
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Change to polar coordinates.
– user10354138
Nov 16 at 23:08
3
How do you integrate $z^nbar{z}^m$ over the region $D$? I mean this makes no sense: $int_D,z^nbar{z}^m,text{d}z$. Did you want to say $$iint_D,z^nbar{z}^m,text{d}z,text{d}bar{z},?$$
– Batominovski
Nov 16 at 23:18