Helmholtz equation Inequality (potential theory)

Question

I was reading the chapter about surface potential. I came across an inequality

$$|e^{ik|x_1-y|}-e^{ik|x_2-y|}|\leq k|x_1-x_2|,$$

$k$ is a complex number here. Since $k$ is a complex number here, I am a bit confused. I am wondering what is the correct way to interpret this ineuqality?

score -1 · Answer 1 · answered May 03 '21 at 08:52

-1

The map $z \mapsto e^z$ maps complex numbers to complex numbers. Hence $e^{ik|x_1-y|}$ is a complex number and

$$|e^{ik|x_1-y|}-e^{ik|x_2-y|}|\leq k|x_1-x_2|$$ is an inequality between complex numbers modulus.

answered May 03 '21 at 08:52

mathcounterexamples.net

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Thank you for your reply. $|\cdot|$ gives real number and $k$ is complex. Is there any assumption involved in comparing real number and complex number? – 000000000 May 03 '21 at 08:57
Left and right sides of the inequality are modulus, i.e. real numbers. – mathcounterexamples.net May 03 '21 at 08:59
Shouldn't we write $|k|$ for $k$ instead? – 000000000 May 03 '21 at 09:00
Well spoted! You're right. This is a mistake in the article. – mathcounterexamples.net May 03 '21 at 09:06
Originally, I thought this is a typo. But I find another resource online. In page 104 (3.15), he used the same notation. https://www.math.kit.edu/ianmip/lehre/sobraum2014s/media/main.pdf – 000000000 May 03 '21 at 09:10
So I am not sure this is a typo or I have some misunderstandings. – 000000000 May 03 '21 at 09:11
I would guess a typo... But not 100% sure. Maybe you can contact the author? – mathcounterexamples.net May 03 '21 at 18:24
yeah maybe, thank you! – 000000000 May 04 '21 at 03:52

Helmholtz equation Inequality (potential theory)

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