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I've been reviewing the contents after quite a while. Now, when I'm reading Kramers-Kronig relations, I see that everywhere it is stated that the complex function have to be analytic in the upper half of the plane.

My question is, will the Kramers-Kronig relation be invalid if the function is analytic everywhere? And if that is so, why the analyticity in the upper half plane is required?

Any link to appropriate texts is also appreciated.

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    $\begingroup$ analytic everywhere implies analytic in upper half plane, so I'm not sure what the confusion is... seems like you're mixing up what is the special case (being analytic everywhere) and what is the general case (analytic in upper half plane) $\endgroup$
    – peek-a-boo
    Aug 30 at 2:00
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The condition of being analytic in the upper half plane is required so that the value of a contour integral around a certain path in the closed upper half plane will equal zero. This contour integral is used in the proof of the Kramers-Konig relations.

If a function is analytic everywhere then it is certainly analytic in the upper half plane, so this condition is met. Conversely, if you only know that the function is analytic in some smaller simply-connected region, then it may be possible to map this region to the upper half plane using a conformal map before applying the K-K relations.

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