# A question on complex analysis

Let $f,g:\mathbb C \to \mathbb C$ be two analytic functions such that $f(z)(g(z)+z^2)=0$ for all $z$ .Then prove that either $f(z)=0$ or $g(z)=-z^2$.

-
The $z^2$ is only a distraction. $h(z)=g(z)+z^2$ is analytic if $g$ is. (I assume that you forgot to mention that $f$ and $g$ are analytic.) What do you know about the zeros of analytic functions? – Jonas Meyer Feb 4 '11 at 15:06
yes ,f and g both are analytic functions . – user6618 Feb 4 '11 at 15:40
Please tell me whether i am correct or not: – user6618 Feb 4 '11 at 15:44
Is Uniqueness theorem applicable to f and this g – user6618 Feb 4 '11 at 15:48
@N Jana: Yes, that's the result you want to apply. Do you see how to apply it? – Jonas Meyer Feb 4 '11 at 15:50

If there exists $a\in\mathbb C$ such that $f(a)\neq 0$, then by continuity there is an open disk at $a$ where $f$ is nonzero. Then $g(z)+z^2$ is zero on this disk, and by the identity theorem for analytic functions, $g(z)+z^2=0$ for all $z\in\mathbb{C}$.
The $z^2$ here adds nothing essential. The obvious generalization is that if a product of analytic functions on a connected open set is identically $0$, then one of the functions is identically zero. (And this means that the ring of analytic functions on a connected open set is an integral domain with pointwise operations.)
If $f=0$ you have done, so can assume this is not the case. Can assume there is a point where both $f$ and $g-z^2$ are both different from zero otherwise you have done. Up to traslation can assume this point is the origin. So there exists integers $h,k$ such that $f=z^hf'$ and $g=z^kg'$ with $f',g'$ different from zero at the origin. then you have $f'(z^{h+k}g'-z^{2+k})=0$. Since $f'$ not zero at the origin this means that locally $z^{h+k}g'-z^{2+k}=0$ from which the thesis follows.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://math.stackexchange.com/questions/20403/a-question-on-complex-analysis", "fetch_time": 1469400153000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2016-30/segments/1469257824185.14/warc/CC-MAIN-20160723071024-00122-ip-10-185-27-174.ec2.internal.warc.gz", "warc_record_offset": 165995371, "warc_record_length": 17585, "token_count": 571, "char_count": 1824, "score": 3.890625, "int_score": 4, "crawl": "CC-MAIN-2016-30", "snapshot_type": "latest", "language": "en", "language_score": 0.8779749870300293, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00051-of-00064.parquet"}