Consider the complex function


First of all, I want to get down with the definition of entire function. Based on Wikipedia, in complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane.

A holomorphic function to me is the same as analytic function.

I don't see the difference, so I am just wondering if we can use Cauchy–Riemann equations to do this exercise.

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    $\begingroup$ It is Cauchy-Riemann equation, not Quotient-Riemann. $\endgroup$ – P Vanchinathan Aug 4 '16 at 0:47
  • $\begingroup$ Correct, sorry XD $\endgroup$ – TheMathNoob Aug 4 '16 at 0:48
  • $\begingroup$ Replace $z$ with $x+iy$ and split the function on the rhs collecting the coefficients of $i$ calling it $v(x,y)$ and the rest as $u(x,y)$. Now apply Cauchy-Riemann. $\endgroup$ – P Vanchinathan Aug 4 '16 at 0:50
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    $\begingroup$ @Nobody, how about "firstiliciously"? :) $\endgroup$ – paul garrett Aug 4 '16 at 0:56
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    $\begingroup$ I have seen it quite often in chat conversations and social networks. I know firstable=first of all. I thought they were equivalent because to be honest English is quite wrong. Some vowels don't correlate with some words. For example, infinite and finite sound different and I haven't found the reason why. $\endgroup$ – TheMathNoob Aug 4 '16 at 1:01

An entire function is defined as a function holomorphic on all of the complex plane. This is correct. Further, yes this is equivalent to the function being analytic. In particular an entire function is given by a power series (and conversely a powerseries converging on all of the complex plane gives an entire function). Thus an entire function is (the same as) a function given by a power series with infinite radius of convergence.

I am not sure what you mean by "Quotient Riemann equations." You could use the Cauchy-Riemmann equation to solve this problem. But this seems a tedious way.

Instead I assume the intended solution is to exploit facts about entire functions. Such as:

  • The sum and product of two entire functions is entire.
  • A polynomial function is entire.
  • The exponential function is entire.
  • The inverse of a non-zero entire function is entire.

With these at hand the result follows quite readily. If you do not know those, try to show them first.


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