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Let $f:\mathbb{C}\to\mathbb{C}$ be a complex valued function of the form $f(x,y)=u(x,y)+iv(x,y)$.

Suppose that $u(x,y)=3x^2y$.


  1. $f$ cannot be holomorphic on $\mathbb{C}$ for any choice of $v$.

  2. $f$ is holomorphic on $\mathbb{C}$ for a suitable choice of $v$.

  3. $f$ is holomorphic on $\mathbb{C}$ for all choices of $v$.

  4. $u$ is not differentiable.

(original image)

well, I have calculated by applying CR equation, getting the option $1$ is correct? could anyone tell me just am I right?thank you.

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I think it would be better (esp. for people with certain browser limitations) to just copy the text of the problem rather than posting an image. – Christopher A. Wong Dec 21 '12 at 5:53
show more work. How did you apply CR. – Thomas Andrews Dec 21 '12 at 5:55
I really do not understand your problem: for people with certain browser limitations, meaning?could you tell me? – La Belle Noiseuse Dec 21 '12 at 5:55
@Kuttus: Regardless of browser limitations, the typed text is far easier to read than the scan you used. A nicer scan would be more acceptable. – Zev Chonoles Dec 21 '12 at 6:01
@ZevChonoles I agree, please pardon me for this time. – La Belle Noiseuse Dec 21 '12 at 6:06
up vote 3 down vote accepted

u and v must satisfy laplace equation if f is a holomorphic function.Any v of course don't
satisfy laplace equation. hence 1 is correct. infact a form of v can be calculated by integrating the CR eequations

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