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February 23rd, 2009, 01:38 PM   #1
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Analytic

show that if f(z) is analytic and real valued in a domain D then f(z) is constant in D.
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February 23rd, 2009, 05:19 PM   #2
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Re: Analytic

Harmonic conjugates are unique to within constants. The harmonic conjugate of the imagimary part of given function is a constant. Since the imag part is zero, the function is a constant.

This uses a powerful result. A direct proof, say from the C-R equations, would be better.
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February 24th, 2009, 05:50 AM   #3
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Re: Analytic

Morera's theorem also helps here.
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February 24th, 2009, 02:09 PM   #4
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Re: Analytic

Quote:
Originally Posted by srw899
show that if f(z) is analytic and real valued in a domain D then f(z) is constant in D.
You need to put some restriction on D. For example if D is the real line and f(z) is a polynomial in z with real coefficients, the statement does not hold.
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February 27th, 2009, 09:10 AM   #5
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Re: Analytic

Good point

Quote:
Originally Posted by mathman
You need to put some restriction on D. For example if D is the real line and f(z) is a polynomial in z with real coefficients, the statement does not hold.
Is not a domain defined to be open? connected? Are these enough, or must we also require boundedness, for example?
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