Meaning of the principal part for an analytic limit (as opposed to an integral)?
I am slightly confused as to the meaning of 'the principle part' of a limit, specifically in relation to the Kramers-Kronig relations as derived here on page 61. The source uses the 'Principal part' in reference to an integral, which I understand as the standard Cauchy Principal part defined as
$P int _{-infty}^{infty} dx f(x) = lim_{epsilon rightarrow 0} [int _{-infty}^{a-epsilon} dx f(x) + int _{a+epsilon}^{infty} dx f(x)]$
for $f(x)$ singular at a.
Now what is causing me confusion is that the source uses the same symbol in reference to a limit
$frac{iomega}{omega ^2 + epsilon ^2} |_{epsilon rightarrow 0} = P frac{i}{omega}$
What does this mean? I don't know how to interpret this. The same symbol is used, but it wouldn't have a definition in terms of the integral... I haven't found a formal definition of 'principal value' for a limit such as this one.
The specific part of the text is here:
EDIT: Another thing I just noticed that I am slightly uneasy about: between equations (7.20) and (7.21) in the image, it seems that
$int domega ' [P(frac{1}{omega - omega '})] v(omega ' )$
(whatever the thing in the square brackets formally means- I still haven't found an answer...) has been turned into:
$P[int domega ' (frac{1}{omega - omega '}) v(omega ' )]$
Of course, the formal meaning of $P(frac{1}{omega - omega '})$ in terms of limits may make absolutely obvious why this is a valid thing to do, but I raise the issue now in case it is not and requires further explanation; hopefully whosoever is able to answer the primary question will be able to add a note on this too.
limits contour-integration cauchy-principal-value
asked 2 days ago
21joanna12
1,0571615
add a comment |
I am slightly confused as to the meaning of 'the principle part' of a limit, specifically in relation to the Kramers-Kronig relations as derived here on page 61. The source uses the 'Principal part' in reference to an integral, which I understand as the standard Cauchy Principal part defined as
$P int _{-infty}^{infty} dx f(x) = lim_{epsilon rightarrow 0} [int _{-infty}^{a-epsilon} dx f(x) + int _{a+epsilon}^{infty} dx f(x)]$
for $f(x)$ singular at a.
Now what is causing me confusion is that the source uses the same symbol in reference to a limit
$frac{iomega}{omega ^2 + epsilon ^2} |_{epsilon rightarrow 0} = P frac{i}{omega}$
What does this mean? I don't know how to interpret this. The same symbol is used, but it wouldn't have a definition in terms of the integral... I haven't found a formal definition of 'principal value' for a limit such as this one.
The specific part of the text is here:
EDIT: Another thing I just noticed that I am slightly uneasy about: between equations (7.20) and (7.21) in the image, it seems that
$int domega ' [P(frac{1}{omega - omega '})] v(omega ' )$
(whatever the thing in the square brackets formally means- I still haven't found an answer...) has been turned into:
$P[int domega ' (frac{1}{omega - omega '}) v(omega ' )]$
Of course, the formal meaning of $P(frac{1}{omega - omega '})$ in terms of limits may make absolutely obvious why this is a valid thing to do, but I raise the issue now in case it is not and requires further explanation; hopefully whosoever is able to answer the primary question will be able to add a note on this too.
limits contour-integration cauchy-principal-value
asked 2 days ago
21joanna12
1,0571615
add a comment |
I am slightly confused as to the meaning of 'the principle part' of a limit, specifically in relation to the Kramers-Kronig relations as derived here on page 61. The source uses the 'Principal part' in reference to an integral, which I understand as the standard Cauchy Principal part defined as
$P int _{-infty}^{infty} dx f(x) = lim_{epsilon rightarrow 0} [int _{-infty}^{a-epsilon} dx f(x) + int _{a+epsilon}^{infty} dx f(x)]$
for $f(x)$ singular at a.
Now what is causing me confusion is that the source uses the same symbol in reference to a limit
$frac{iomega}{omega ^2 + epsilon ^2} |_{epsilon rightarrow 0} = P frac{i}{omega}$
What does this mean? I don't know how to interpret this. The same symbol is used, but it wouldn't have a definition in terms of the integral... I haven't found a formal definition of 'principal value' for a limit such as this one.
The specific part of the text is here:
EDIT: Another thing I just noticed that I am slightly uneasy about: between equations (7.20) and (7.21) in the image, it seems that
$int domega ' [P(frac{1}{omega - omega '})] v(omega ' )$
(whatever the thing in the square brackets formally means- I still haven't found an answer...) has been turned into:
$P[int domega ' (frac{1}{omega - omega '}) v(omega ' )]$
Of course, the formal meaning of $P(frac{1}{omega - omega '})$ in terms of limits may make absolutely obvious why this is a valid thing to do, but I raise the issue now in case it is not and requires further explanation; hopefully whosoever is able to answer the primary question will be able to add a note on this too.
limits contour-integration cauchy-principal-value
asked 2 days ago
21joanna12
1,0571615
I am slightly confused as to the meaning of 'the principle part' of a limit, specifically in relation to the Kramers-Kronig relations as derived here on page 61. The source uses the 'Principal part' in reference to an integral, which I understand as the standard Cauchy Principal part defined as
$P int _{-infty}^{infty} dx f(x) = lim_{epsilon rightarrow 0} [int _{-infty}^{a-epsilon} dx f(x) + int _{a+epsilon}^{infty} dx f(x)]$
for $f(x)$ singular at a.
Now what is causing me confusion is that the source uses the same symbol in reference to a limit
$frac{iomega}{omega ^2 + epsilon ^2} |_{epsilon rightarrow 0} = P frac{i}{omega}$
What does this mean? I don't know how to interpret this. The same symbol is used, but it wouldn't have a definition in terms of the integral... I haven't found a formal definition of 'principal value' for a limit such as this one.
The specific part of the text is here:
EDIT: Another thing I just noticed that I am slightly uneasy about: between equations (7.20) and (7.21) in the image, it seems that
$int domega ' [P(frac{1}{omega - omega '})] v(omega ' )$
(whatever the thing in the square brackets formally means- I still haven't found an answer...) has been turned into:
$P[int domega ' (frac{1}{omega - omega '}) v(omega ' )]$
Of course, the formal meaning of $P(frac{1}{omega - omega '})$ in terms of limits may make absolutely obvious why this is a valid thing to do, but I raise the issue now in case it is not and requires further explanation; hopefully whosoever is able to answer the primary question will be able to add a note on this too.
limits contour-integration cauchy-principal-value
limits contour-integration cauchy-principal-value
asked 2 days ago
21joanna12
1,0571615
asked 2 days ago
21joanna12
1,0571615
edited 2 days ago
asked 2 days ago
21joanna12
1,0571615
asked 2 days ago
21joanna12
1,0571615
asked 2 days ago
21joanna12
1,0571615
1,0571615
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