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Is this Function Holomorphic (Riesz-Herglotz Representation)?
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Let $g(z)$ be a holomorphic function on the closed unit disc $overline mathbb D$
Is the function $ g_1(w) = 1/2pi int_0^2pi operatornameRe(g(e^itheta))(e^itheta + w)/(e^itheta - w) , dtheta $ holomorphic on the open unit disc $ mathbb D$ ?
From the context (see below) I presume that it is but I don't see how to prove this.
Context
This is part of a proof here on p.29 :
https://su.diva-portal.org/smash/get/diva2:1069992/FULLTEXT01.pdf
to establish the Riesz-Herglotz representation of a holomorphic function on $mathbb D$ with a non-negative real part.
I follow the proof up to establishing an expression for $w in mathbb D,$
$$ g(w) = frac 1 2pi int_0^2pi g(e^itheta) operatornameRe left( frace^itheta + we^itheta - wright) , dtheta $$
I can also see that $operatornameRe(g(w)) = operatornameRe(g_1(w))$ so if $g_1$ is holomorphic one can then say that $g, g_1$ differ only by a complex constant.
Then I could continue to follow the proof. Help would be appreciated.
complex-analysis complex-integration
edited Jul 16 at 12:00
Michael Hardy
204k23186463
asked Jul 16 at 11:29
Tom Collinge
4,331932
add a comment |Â
up vote
0
down vote
favorite
Let $g(z)$ be a holomorphic function on the closed unit disc $overline mathbb D$
Is the function $ g_1(w) = 1/2pi int_0^2pi operatornameRe(g(e^itheta))(e^itheta + w)/(e^itheta - w) , dtheta $ holomorphic on the open unit disc $ mathbb D$ ?
From the context (see below) I presume that it is but I don't see how to prove this.
Context
This is part of a proof here on p.29 :
https://su.diva-portal.org/smash/get/diva2:1069992/FULLTEXT01.pdf
to establish the Riesz-Herglotz representation of a holomorphic function on $mathbb D$ with a non-negative real part.
I follow the proof up to establishing an expression for $w in mathbb D,$
$$ g(w) = frac 1 2pi int_0^2pi g(e^itheta) operatornameRe left( frace^itheta + we^itheta - wright) , dtheta $$
I can also see that $operatornameRe(g(w)) = operatornameRe(g_1(w))$ so if $g_1$ is holomorphic one can then say that $g, g_1$ differ only by a complex constant.
Then I could continue to follow the proof. Help would be appreciated.
complex-analysis complex-integration
edited Jul 16 at 12:00
Michael Hardy
204k23186463
asked Jul 16 at 11:29
Tom Collinge
4,331932
$dfracpartialpartialbarwg_1(w)=0$
– Nosrati
Jul 16 at 13:49
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $g(z)$ be a holomorphic function on the closed unit disc $overline mathbb D$
Is the function $ g_1(w) = 1/2pi int_0^2pi operatornameRe(g(e^itheta))(e^itheta + w)/(e^itheta - w) , dtheta $ holomorphic on the open unit disc $ mathbb D$ ?
From the context (see below) I presume that it is but I don't see how to prove this.
Context
This is part of a proof here on p.29 :
https://su.diva-portal.org/smash/get/diva2:1069992/FULLTEXT01.pdf
to establish the Riesz-Herglotz representation of a holomorphic function on $mathbb D$ with a non-negative real part.
I follow the proof up to establishing an expression for $w in mathbb D,$
$$ g(w) = frac 1 2pi int_0^2pi g(e^itheta) operatornameRe left( frace^itheta + we^itheta - wright) , dtheta $$
I can also see that $operatornameRe(g(w)) = operatornameRe(g_1(w))$ so if $g_1$ is holomorphic one can then say that $g, g_1$ differ only by a complex constant.
Then I could continue to follow the proof. Help would be appreciated.
complex-analysis complex-integration
edited Jul 16 at 12:00
Michael Hardy
204k23186463
asked Jul 16 at 11:29
Tom Collinge
4,331932
Let $g(z)$ be a holomorphic function on the closed unit disc $overline mathbb D$
Is the function $ g_1(w) = 1/2pi int_0^2pi operatornameRe(g(e^itheta))(e^itheta + w)/(e^itheta - w) , dtheta $ holomorphic on the open unit disc $ mathbb D$ ?
From the context (see below) I presume that it is but I don't see how to prove this.
Context
This is part of a proof here on p.29 :
https://su.diva-portal.org/smash/get/diva2:1069992/FULLTEXT01.pdf
to establish the Riesz-Herglotz representation of a holomorphic function on $mathbb D$ with a non-negative real part.
I follow the proof up to establishing an expression for $w in mathbb D,$
$$ g(w) = frac 1 2pi int_0^2pi g(e^itheta) operatornameRe left( frace^itheta + we^itheta - wright) , dtheta $$
I can also see that $operatornameRe(g(w)) = operatornameRe(g_1(w))$ so if $g_1$ is holomorphic one can then say that $g, g_1$ differ only by a complex constant.
Then I could continue to follow the proof. Help would be appreciated.
complex-analysis complex-integration
edited Jul 16 at 12:00
Michael Hardy
204k23186463
asked Jul 16 at 11:29
Tom Collinge
4,331932
edited Jul 16 at 12:00
Michael Hardy
204k23186463
edited Jul 16 at 12:00
Michael Hardy
204k23186463
edited Jul 16 at 12:00
Michael Hardy
204k23186463
204k23186463
asked Jul 16 at 11:29
Tom Collinge
4,331932
asked Jul 16 at 11:29
Tom Collinge
4,331932
asked Jul 16 at 11:29
Tom Collinge
4,331932
4,331932
$dfracpartialpartialbarwg_1(w)=0$
– Nosrati
Jul 16 at 13:49
add a comment |Â
$dfracpartialpartialbarwg_1(w)=0$
– Nosrati
Jul 16 at 13:49
$dfracpartialpartialbarwg_1(w)=0$
– Nosrati
Jul 16 at 13:49
$dfracpartialpartialbarwg_1(w)=0$
– Nosrati
Jul 16 at 13:49
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
1
down vote
Theorem 7 in Rudin's RCA (it is actually Theorem 10.7 in my Indian edition of the book) immediately tells you that $g_1$ is holomorphic in $D$. Here are the details: let $g_2(w)=frac 1 2pi int_0^2pi frac 1 e^itheta -w dmu (theta )$ where $dmu (theta ) =Re (g(e^itheta ))e^itheta , dtheta$ and $g_3(w)=frac 1 2pi int_0^2pi frac 1 e^itheta -w dnu (theta )$ where $dnu (theta ) =Re (g(e^itheta )) , dtheta$. Then $g_1(w)=g_2(w)+wg_3(w)$. $g_2$ and $g_3$ are both holomorphic by the theorem in Rudin's book.
answered Jul 16 at 11:46
Kavi Rama Murthy
21k2830
Thanks: I have the INTERNATIONAL EDITION 1987. Could you give a little more detail to help me find it - chapter title, page number, opening words of the theorem ?
– Tom Collinge
Jul 16 at 11:52
@TomCollinge It is in the chapter on Elementary Properties of Holomorphic Functions, bottom of 4-th page in that chapter. The theorem immediately precedes 'Integration over paths". I hope you can locate it. Otherwise, I will type out the exact statement of the theorem.
– Kavi Rama Murthy
Jul 16 at 11:56
Thanks. 10.7 in my book is "Suppose $mu$ is a complex (finite) measure .....", but I don;t see how to apply it. Note: I know that $g$ is holomorphic, it's $g_1$ that gives me a problem.
– Tom Collinge
Jul 16 at 11:59
@TomCollinge I have provided the details now.
– Kavi Rama Murthy
Jul 16 at 23:24
Thanks. That helps a lot.
– Tom Collinge
Jul 17 at 7:27
add a comment |Â
up vote
1
down vote
The answer goes along the lines of the answer to the question Analytic functions defined by integrals.
In our case, we begin by making two easy observations:
(1). The function $theta mapsto Re[g(e^i theta)]$ is bounded for $theta in [0,2pi)$. This is due to the assumption that $g$ is holomorphic in $overlinemathbbD$.
(2). The function $w mapsto (e^i theta + w)/(e^i theta - w)$ is holomorphic in $mathbbD$ for any fixed $theta in [0,2pi)$.
From (1) and (2), we quickly infer the following:
(3). Given a compact set $U subseteq mathbbD$, there exists a constant $C in mathbbR$, such that for any $w in U$, it holds that
$$int_0^2pileft|Re[g(e^i theta)]frace^i theta + we^i theta - wright|mathrmdtheta leq C.$$
Morera's theorem now gives that $g_1$ is holomorphic in $mathbbD$. Indeed, if $T subseteq mathbbD$ is any triangle, it holds that
beginalign
oint_Tg_1(w)mathrmdw & = frac12pioint_Tint_0^2piRe[g(e^i theta)]frace^i theta + we^i theta - wmathrmdtheta:mathrmdw \
& = frac12piint_0^2piRe[g(e^i theta)]underbraceoint_Tfrace^i theta + we^i theta - wmathrmdw_= 0:mathrmdue~to~ (2):mathrmdtheta \
& = 0.
endalign
Here, it was valid to change the order of integration as (3) allows us to apply Fubini's theorem and change the order of integration.
answered Jul 17 at 7:41
nmbcktt
111
Very helpful. Nicely formatted.
– Tom Collinge
Jul 17 at 7:59
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Theorem 7 in Rudin's RCA (it is actually Theorem 10.7 in my Indian edition of the book) immediately tells you that $g_1$ is holomorphic in $D$. Here are the details: let $g_2(w)=frac 1 2pi int_0^2pi frac 1 e^itheta -w dmu (theta )$ where $dmu (theta ) =Re (g(e^itheta ))e^itheta , dtheta$ and $g_3(w)=frac 1 2pi int_0^2pi frac 1 e^itheta -w dnu (theta )$ where $dnu (theta ) =Re (g(e^itheta )) , dtheta$. Then $g_1(w)=g_2(w)+wg_3(w)$. $g_2$ and $g_3$ are both holomorphic by the theorem in Rudin's book.
answered Jul 16 at 11:46
Kavi Rama Murthy
21k2830
Thanks: I have the INTERNATIONAL EDITION 1987. Could you give a little more detail to help me find it - chapter title, page number, opening words of the theorem ?
– Tom Collinge
Jul 16 at 11:52
@TomCollinge It is in the chapter on Elementary Properties of Holomorphic Functions, bottom of 4-th page in that chapter. The theorem immediately precedes 'Integration over paths". I hope you can locate it. Otherwise, I will type out the exact statement of the theorem.
– Kavi Rama Murthy
Jul 16 at 11:56
Thanks. 10.7 in my book is "Suppose $mu$ is a complex (finite) measure .....", but I don;t see how to apply it. Note: I know that $g$ is holomorphic, it's $g_1$ that gives me a problem.
– Tom Collinge
Jul 16 at 11:59
@TomCollinge I have provided the details now.
– Kavi Rama Murthy
Jul 16 at 23:24
Thanks. That helps a lot.
– Tom Collinge
Jul 17 at 7:27
add a comment |Â
up vote
1
down vote
Theorem 7 in Rudin's RCA (it is actually Theorem 10.7 in my Indian edition of the book) immediately tells you that $g_1$ is holomorphic in $D$. Here are the details: let $g_2(w)=frac 1 2pi int_0^2pi frac 1 e^itheta -w dmu (theta )$ where $dmu (theta ) =Re (g(e^itheta ))e^itheta , dtheta$ and $g_3(w)=frac 1 2pi int_0^2pi frac 1 e^itheta -w dnu (theta )$ where $dnu (theta ) =Re (g(e^itheta )) , dtheta$. Then $g_1(w)=g_2(w)+wg_3(w)$. $g_2$ and $g_3$ are both holomorphic by the theorem in Rudin's book.
answered Jul 16 at 11:46
Kavi Rama Murthy
21k2830
Thanks: I have the INTERNATIONAL EDITION 1987. Could you give a little more detail to help me find it - chapter title, page number, opening words of the theorem ?
– Tom Collinge
Jul 16 at 11:52
@TomCollinge It is in the chapter on Elementary Properties of Holomorphic Functions, bottom of 4-th page in that chapter. The theorem immediately precedes 'Integration over paths". I hope you can locate it. Otherwise, I will type out the exact statement of the theorem.
– Kavi Rama Murthy
Jul 16 at 11:56
Thanks. 10.7 in my book is "Suppose $mu$ is a complex (finite) measure .....", but I don;t see how to apply it. Note: I know that $g$ is holomorphic, it's $g_1$ that gives me a problem.
– Tom Collinge
Jul 16 at 11:59
@TomCollinge I have provided the details now.
– Kavi Rama Murthy
Jul 16 at 23:24
Thanks. That helps a lot.
– Tom Collinge
Jul 17 at 7:27
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Theorem 7 in Rudin's RCA (it is actually Theorem 10.7 in my Indian edition of the book) immediately tells you that $g_1$ is holomorphic in $D$. Here are the details: let $g_2(w)=frac 1 2pi int_0^2pi frac 1 e^itheta -w dmu (theta )$ where $dmu (theta ) =Re (g(e^itheta ))e^itheta , dtheta$ and $g_3(w)=frac 1 2pi int_0^2pi frac 1 e^itheta -w dnu (theta )$ where $dnu (theta ) =Re (g(e^itheta )) , dtheta$. Then $g_1(w)=g_2(w)+wg_3(w)$. $g_2$ and $g_3$ are both holomorphic by the theorem in Rudin's book.
answered Jul 16 at 11:46
Kavi Rama Murthy
21k2830
Theorem 7 in Rudin's RCA (it is actually Theorem 10.7 in my Indian edition of the book) immediately tells you that $g_1$ is holomorphic in $D$. Here are the details: let $g_2(w)=frac 1 2pi int_0^2pi frac 1 e^itheta -w dmu (theta )$ where $dmu (theta ) =Re (g(e^itheta ))e^itheta , dtheta$ and $g_3(w)=frac 1 2pi int_0^2pi frac 1 e^itheta -w dnu (theta )$ where $dnu (theta ) =Re (g(e^itheta )) , dtheta$. Then $g_1(w)=g_2(w)+wg_3(w)$. $g_2$ and $g_3$ are both holomorphic by the theorem in Rudin's book.
answered Jul 16 at 11:46
Kavi Rama Murthy
21k2830
edited Jul 16 at 23:23
answered Jul 16 at 11:46
Kavi Rama Murthy
21k2830
answered Jul 16 at 11:46
Kavi Rama Murthy
21k2830
answered Jul 16 at 11:46
Kavi Rama Murthy
21k2830
21k2830
Thanks: I have the INTERNATIONAL EDITION 1987. Could you give a little more detail to help me find it - chapter title, page number, opening words of the theorem ?
– Tom Collinge
Jul 16 at 11:52
@TomCollinge It is in the chapter on Elementary Properties of Holomorphic Functions, bottom of 4-th page in that chapter. The theorem immediately precedes 'Integration over paths". I hope you can locate it. Otherwise, I will type out the exact statement of the theorem.
– Kavi Rama Murthy
Jul 16 at 11:56
Thanks. 10.7 in my book is "Suppose $mu$ is a complex (finite) measure .....", but I don;t see how to apply it. Note: I know that $g$ is holomorphic, it's $g_1$ that gives me a problem.
– Tom Collinge
Jul 16 at 11:59
@TomCollinge I have provided the details now.
– Kavi Rama Murthy
Jul 16 at 23:24
Thanks. That helps a lot.
– Tom Collinge
Jul 17 at 7:27
add a comment |Â
Thanks: I have the INTERNATIONAL EDITION 1987. Could you give a little more detail to help me find it - chapter title, page number, opening words of the theorem ?
– Tom Collinge
Jul 16 at 11:52
@TomCollinge It is in the chapter on Elementary Properties of Holomorphic Functions, bottom of 4-th page in that chapter. The theorem immediately precedes 'Integration over paths". I hope you can locate it. Otherwise, I will type out the exact statement of the theorem.
– Kavi Rama Murthy
Jul 16 at 11:56
Thanks. 10.7 in my book is "Suppose $mu$ is a complex (finite) measure .....", but I don;t see how to apply it. Note: I know that $g$ is holomorphic, it's $g_1$ that gives me a problem.
– Tom Collinge
Jul 16 at 11:59
@TomCollinge I have provided the details now.
– Kavi Rama Murthy
Jul 16 at 23:24
Thanks. That helps a lot.
– Tom Collinge
Jul 17 at 7:27
Thanks: I have the INTERNATIONAL EDITION 1987. Could you give a little more detail to help me find it - chapter title, page number, opening words of the theorem ?
– Tom Collinge
Jul 16 at 11:52
Thanks: I have the INTERNATIONAL EDITION 1987. Could you give a little more detail to help me find it - chapter title, page number, opening words of the theorem ?
– Tom Collinge
Jul 16 at 11:52
@TomCollinge It is in the chapter on Elementary Properties of Holomorphic Functions, bottom of 4-th page in that chapter. The theorem immediately precedes 'Integration over paths". I hope you can locate it. Otherwise, I will type out the exact statement of the theorem.
– Kavi Rama Murthy
Jul 16 at 11:56
@TomCollinge It is in the chapter on Elementary Properties of Holomorphic Functions, bottom of 4-th page in that chapter. The theorem immediately precedes 'Integration over paths". I hope you can locate it. Otherwise, I will type out the exact statement of the theorem.
– Kavi Rama Murthy
Jul 16 at 11:56
Thanks. 10.7 in my book is "Suppose $mu$ is a complex (finite) measure .....", but I don;t see how to apply it. Note: I know that $g$ is holomorphic, it's $g_1$ that gives me a problem.
– Tom Collinge
Jul 16 at 11:59
Thanks. 10.7 in my book is "Suppose $mu$ is a complex (finite) measure .....", but I don;t see how to apply it. Note: I know that $g$ is holomorphic, it's $g_1$ that gives me a problem.
– Tom Collinge
Jul 16 at 11:59
@TomCollinge I have provided the details now.
– Kavi Rama Murthy
Jul 16 at 23:24
@TomCollinge I have provided the details now.
– Kavi Rama Murthy
Jul 16 at 23:24
Thanks. That helps a lot.
– Tom Collinge
Jul 17 at 7:27
Thanks. That helps a lot.
– Tom Collinge
Jul 17 at 7:27
add a comment |Â
up vote
1
down vote
The answer goes along the lines of the answer to the question Analytic functions defined by integrals.
In our case, we begin by making two easy observations:
(1). The function $theta mapsto Re[g(e^i theta)]$ is bounded for $theta in [0,2pi)$. This is due to the assumption that $g$ is holomorphic in $overlinemathbbD$.
(2). The function $w mapsto (e^i theta + w)/(e^i theta - w)$ is holomorphic in $mathbbD$ for any fixed $theta in [0,2pi)$.
From (1) and (2), we quickly infer the following:
(3). Given a compact set $U subseteq mathbbD$, there exists a constant $C in mathbbR$, such that for any $w in U$, it holds that
$$int_0^2pileft|Re[g(e^i theta)]frace^i theta + we^i theta - wright|mathrmdtheta leq C.$$
Morera's theorem now gives that $g_1$ is holomorphic in $mathbbD$. Indeed, if $T subseteq mathbbD$ is any triangle, it holds that
beginalign
oint_Tg_1(w)mathrmdw & = frac12pioint_Tint_0^2piRe[g(e^i theta)]frace^i theta + we^i theta - wmathrmdtheta:mathrmdw \
& = frac12piint_0^2piRe[g(e^i theta)]underbraceoint_Tfrace^i theta + we^i theta - wmathrmdw_= 0:mathrmdue~to~ (2):mathrmdtheta \
& = 0.
endalign
Here, it was valid to change the order of integration as (3) allows us to apply Fubini's theorem and change the order of integration.
answered Jul 17 at 7:41
nmbcktt
111
Very helpful. Nicely formatted.
– Tom Collinge
Jul 17 at 7:59
add a comment |Â
up vote
1
down vote
The answer goes along the lines of the answer to the question Analytic functions defined by integrals.
In our case, we begin by making two easy observations:
(1). The function $theta mapsto Re[g(e^i theta)]$ is bounded for $theta in [0,2pi)$. This is due to the assumption that $g$ is holomorphic in $overlinemathbbD$.
(2). The function $w mapsto (e^i theta + w)/(e^i theta - w)$ is holomorphic in $mathbbD$ for any fixed $theta in [0,2pi)$.
From (1) and (2), we quickly infer the following:
(3). Given a compact set $U subseteq mathbbD$, there exists a constant $C in mathbbR$, such that for any $w in U$, it holds that
$$int_0^2pileft|Re[g(e^i theta)]frace^i theta + we^i theta - wright|mathrmdtheta leq C.$$
Morera's theorem now gives that $g_1$ is holomorphic in $mathbbD$. Indeed, if $T subseteq mathbbD$ is any triangle, it holds that
beginalign
oint_Tg_1(w)mathrmdw & = frac12pioint_Tint_0^2piRe[g(e^i theta)]frace^i theta + we^i theta - wmathrmdtheta:mathrmdw \
& = frac12piint_0^2piRe[g(e^i theta)]underbraceoint_Tfrace^i theta + we^i theta - wmathrmdw_= 0:mathrmdue~to~ (2):mathrmdtheta \
& = 0.
endalign
Here, it was valid to change the order of integration as (3) allows us to apply Fubini's theorem and change the order of integration.
answered Jul 17 at 7:41
nmbcktt
111
Very helpful. Nicely formatted.
– Tom Collinge
Jul 17 at 7:59
add a comment |Â
up vote
1
down vote
up vote
1
down vote
The answer goes along the lines of the answer to the question Analytic functions defined by integrals.
In our case, we begin by making two easy observations:
(1). The function $theta mapsto Re[g(e^i theta)]$ is bounded for $theta in [0,2pi)$. This is due to the assumption that $g$ is holomorphic in $overlinemathbbD$.
(2). The function $w mapsto (e^i theta + w)/(e^i theta - w)$ is holomorphic in $mathbbD$ for any fixed $theta in [0,2pi)$.
From (1) and (2), we quickly infer the following:
(3). Given a compact set $U subseteq mathbbD$, there exists a constant $C in mathbbR$, such that for any $w in U$, it holds that
$$int_0^2pileft|Re[g(e^i theta)]frace^i theta + we^i theta - wright|mathrmdtheta leq C.$$
Morera's theorem now gives that $g_1$ is holomorphic in $mathbbD$. Indeed, if $T subseteq mathbbD$ is any triangle, it holds that
beginalign
oint_Tg_1(w)mathrmdw & = frac12pioint_Tint_0^2piRe[g(e^i theta)]frace^i theta + we^i theta - wmathrmdtheta:mathrmdw \
& = frac12piint_0^2piRe[g(e^i theta)]underbraceoint_Tfrace^i theta + we^i theta - wmathrmdw_= 0:mathrmdue~to~ (2):mathrmdtheta \
& = 0.
endalign
Here, it was valid to change the order of integration as (3) allows us to apply Fubini's theorem and change the order of integration.
answered Jul 17 at 7:41
nmbcktt
111
The answer goes along the lines of the answer to the question Analytic functions defined by integrals.
In our case, we begin by making two easy observations:
(1). The function $theta mapsto Re[g(e^i theta)]$ is bounded for $theta in [0,2pi)$. This is due to the assumption that $g$ is holomorphic in $overlinemathbbD$.
(2). The function $w mapsto (e^i theta + w)/(e^i theta - w)$ is holomorphic in $mathbbD$ for any fixed $theta in [0,2pi)$.
From (1) and (2), we quickly infer the following:
(3). Given a compact set $U subseteq mathbbD$, there exists a constant $C in mathbbR$, such that for any $w in U$, it holds that
$$int_0^2pileft|Re[g(e^i theta)]frace^i theta + we^i theta - wright|mathrmdtheta leq C.$$
Morera's theorem now gives that $g_1$ is holomorphic in $mathbbD$. Indeed, if $T subseteq mathbbD$ is any triangle, it holds that
beginalign
oint_Tg_1(w)mathrmdw & = frac12pioint_Tint_0^2piRe[g(e^i theta)]frace^i theta + we^i theta - wmathrmdtheta:mathrmdw \
& = frac12piint_0^2piRe[g(e^i theta)]underbraceoint_Tfrace^i theta + we^i theta - wmathrmdw_= 0:mathrmdue~to~ (2):mathrmdtheta \
& = 0.
endalign
Here, it was valid to change the order of integration as (3) allows us to apply Fubini's theorem and change the order of integration.
answered Jul 17 at 7:41
nmbcktt
111
answered Jul 17 at 7:41
nmbcktt
111
answered Jul 17 at 7:41
nmbcktt
111
answered Jul 17 at 7:41
nmbcktt
111
111
Very helpful. Nicely formatted.
– Tom Collinge
Jul 17 at 7:59
add a comment |Â
Very helpful. Nicely formatted.
– Tom Collinge
Jul 17 at 7:59
Very helpful. Nicely formatted.
– Tom Collinge
Jul 17 at 7:59
Very helpful. Nicely formatted.
– Tom Collinge
Jul 17 at 7:59
add a comment |Â
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$dfracpartialpartialbarwg_1(w)=0$
– Nosrati
Jul 16 at 13:49