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bmanbs2
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Homework Statement
Suppose [tex]f[/tex] and [tex]g[/tex] are analytic on a bounded domain [tex]D[/tex] and continuous on the domain's boundary [tex]B[/tex].
Also, [tex]Re\left(f\right) = Re\left(g\right)[/tex] on [tex]B[/tex].
Show that [tex]f = g + ia[/tex], where [tex]a[/tex] is a real number.
Homework Equations
The maximum modulus principle states that [tex]Re\left(f\right)[/tex] and [tex]Re\left(g\right)[/tex] have no local minima or maxima on [tex]D[/tex], and that the absolute values of [tex]Re\left(f\right)[/tex], [tex]Re\left(g\right)[/tex], [tex]f[/tex], and [tex]g[/tex] have maximums on [tex]B[/tex].
The Attempt at a Solution
I'm not sure how to show [tex]Re\left(f\right) = Re\left(g\right)[/tex] across the entire domain.
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