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Full disclosure: I'm taking my first complex analysis course as a graduate student and the title of my question looks like a dumb question to me.

In any case, there's a problem in my book that deals with a sequence of "holomorphic polynomials" converging to a "holomorphic polynomial". Is this just a redundancy or is there some weird world where certain polynomials aren't (complex) differentiable?

eeeeeeeeee
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2 Answers2

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The sum and product of two holomorphic functions is a holomorphic function.

As constant functions and the identity function $z\in\mathbb C\mapsto z\in\mathbb C$ are holomorphic, it follows that all polynomials are holomorphic functions.

Mariano Suárez-Álvarez
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  • Yeah, that's what I thought. It reminds me of my first Spanish class in which I learned that I didn't need to say "Yo tengo..." since "Tengo..." was sufficient, and that the "Yo" maybe was just overemphasizing the statement. – eeeeeeeeee Feb 22 '15 at 20:01
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    Conceivably one could talk about polynomials in $z$ and $\bar z$, so it really depends on how you define "polynomial". – Santiago Canez Feb 22 '15 at 20:02
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    @SantiagoCanez, You could conceivably define polynomials to include strawberries, too. But the standard definition excludes that. – Mariano Suárez-Álvarez Feb 22 '15 at 20:07
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    @MarianoSuárez-Alvarez, I disagree. It is not beyond the realm of reason to think that the book in question considers "polynomial" to simply mean a polynomial in $x$ and $y$. – Santiago Canez Feb 22 '15 at 20:10
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    @SantiagoCanez One can think all sort of things, and it is rather pointless to speculate on all of them. If you want confirmation, I suggest you add a comment on the question askng what exactly is a polynomial in the context of the book. As you presumably can tell, I cannot help you on that matter. – Mariano Suárez-Álvarez Feb 22 '15 at 20:13
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In some contexts, it makes sense to talk about polynomials in $x$ and $y$ (where $z=x+iy$) even in a complex analysis course. Alternatively, we can take polynomials in $z$ and $\bar z$ which turns out to be the same.

If this is the case in your textbook, it should have been made clear.

The more common convention, especially in introductory courses, is to use "polynomial" for polynomials in $z$ (or elements in $\mathbb{C}[z]$ if you prefer a more algebraic language). Such functions are indeed automatically holomorphic.

mrf
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