- #1

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- TL;DR Summary
- utilizing even and odd functions as the punchline

I am reading a proof in Feedback Systems by Astrom, for the Bode Sensitivity Integral, pg 339. I am stuck on a specific part of the proof.

He is evaluating an integral along a contour which makes up the imaginary axis. He has the following:

$$ -i\int_{-iR}^{iR} log(S(iw))dw=-i\int_{-iR}^{iR}log\left | S(iw) \right |dw -i\int_{-iR}^{iR}\angle S(iw)dw=-2i\int_{0}^{iR}log\left | S(iw) \right |dw $$

Taking a look at the second, or middle expression above, he makes the claim the second term, or imaginary part of the logarithm, is odd, while the first term is even.

Given that the angle on the upper part of the y-ordinate is pi/2 and the lower part is -pi/2, the fact that the second term is odd seems obvious, so this integral vanishes.

I would like to show that the first term is even for any complex function.

This is easy to show for a simple example like $$ f(z) = Re^{i \theta} $$

I am struggling to show it for any function $$ f(z)=f(Re^{i \theta}) $$

I wrote a MATLAB script that allows me to put in any f(z) I can think of and its always true to the claim. I would just like to "prove" it.

He is evaluating an integral along a contour which makes up the imaginary axis. He has the following:

$$ -i\int_{-iR}^{iR} log(S(iw))dw=-i\int_{-iR}^{iR}log\left | S(iw) \right |dw -i\int_{-iR}^{iR}\angle S(iw)dw=-2i\int_{0}^{iR}log\left | S(iw) \right |dw $$

Taking a look at the second, or middle expression above, he makes the claim the second term, or imaginary part of the logarithm, is odd, while the first term is even.

Given that the angle on the upper part of the y-ordinate is pi/2 and the lower part is -pi/2, the fact that the second term is odd seems obvious, so this integral vanishes.

I would like to show that the first term is even for any complex function.

This is easy to show for a simple example like $$ f(z) = Re^{i \theta} $$

I am struggling to show it for any function $$ f(z)=f(Re^{i \theta}) $$

I wrote a MATLAB script that allows me to put in any f(z) I can think of and its always true to the claim. I would just like to "prove" it.