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While learning some consequences of the Cauchy-Riemann theorem. we learned that

An analytic function with constant imaginary (or real) part is constant.

and in addition,

Sum of analytic functions is analytic.

Given the second proposition, I understand that the function $f(x+iy)=x+2i$ is analytic. On the other hand, from the first proposition, it must be constant, but it is not (it depends on $x$, which is not a constant.

How can this contradiction be settled? What am I getting wrong here?

Thank you

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    $\begingroup$ The function $f(x+iy) = x + 2i$ is not analytic! That is because the function $x+iy \mapsto x$ is not analytic. Why did you think it is? Try computing the complex derivative of the real part function (or the imaginary part function). Don't confuse real-analytic and complex-analytic functions. The CR equations are related to complex-analytic functions. $\endgroup$
    – KCd
    Nov 7, 2021 at 7:02
  • $\begingroup$ @KCd: thank you! I completely missed it unfortunately :( $\endgroup$
    – Dr. John
    Nov 7, 2021 at 16:10

1 Answer 1

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The function you considered here is $x+2i$. Now, your logic is that the function $``x"$ is analytic and the function $``2i"$ is analytic (although constant). Now, you argument is that sum of two analytic function is constant.

What gone wrong here is the assumption that your first function $``x"$ is analytic. The function $g(x+iy)=x$ is not analytic at all, since its imaginary part is zero, i.e., constant. By your first result, if $g$ has to be analytic, then it must be constant, as its imaginary part is constant. However, g is clearly not a constant function.

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