# Integral question: zeroes of the primitive.

Let $z$ be a complex number. Let $f(z)$ be an elementary function but not a polynomial. Let its integral $F(z)$ be impossible to express in elementary functions. If we define $F(z)$ as $\int$ from $A$ to $z$ (in a straight path from $A$ to $z$), how do we know if there is a finite complex $A$? What if there are multiple values where $F(z)$ = $0$; What $A$ to choose? How to compute $A$? What $A$ makes the computation of the integral the easiest? Is it the $A$ closest to $z$?

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Could you please take a little time to expand your question? Some details are unclear. For example, what is $A$, is it a complex number? When you integrate from $A$ to $z$, which path do you integrate along? Does $f$ have poles and if so, what kind? Finally, if you want to define $F$ as an integral then it's custom to use a dummy variable: $$\ln x := \int_1^x \frac{dt}{t} \, .$$ – Fly by Night Sep 9 '12 at 17:56
It usually makes more sense to start with some convenient $A$ and take the antiderivative (in some neighbourhood of $A$) that is $0$ at $A$. – Robert Israel Sep 9 '12 at 18:20
@Robert Israel : I think what you meant is we work with the integration constant and/or integrate f(z) + C_2 such that we can find an antiderivative at A that is 0 whereever we want (and then pick a convenient A and/or place). – mick Sep 9 '12 at 18:29
@Fly by Night : i edited it slightly. Im aware of the dummy variable. – mick Sep 9 '12 at 18:30

## 1 Answer

If $f$ is an entire function that is not a polynomial and $F$ is an antiderivative of it, the little Picard theorem says $F$ takes on every complex value with at most one exception. That exception could be $0$, in which case $F = \exp(G)$ for some entire function $G$. Offhand I couldn't think of an example of this where $f$ is elementary but $F$ is not.

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Dear Robert , i appreciate your effort. But i must say this has learned me absolutely nothing : 1) f entire and elementary is very restrictive 2) i was well aware of that for many many years 3) my question is far from answered. As summary , id say this would make a good comment for other people to read , but for me its trivial and very far from an answer. But dont worry i wont downvote since its clearly true. Though i did expect at least an example of f entire where F = exp(G) considering your reputation and skills :) – mick Sep 9 '12 at 19:08
Sorry to disappoint you. – Robert Israel Sep 9 '12 at 22:25
@mick: I'm sure what you meant was "@RobertIsrael: thank you for your answer. Would you please provide some more details along these lines?" There is no way that someone answering your question knows how much you know, and given this, Robert's answer is quite reasonable. After politely explaining what you have tried, I'm sure that Robert would be inclined to expand his answer. – robjohn Sep 10 '12 at 14:41
Thank you for your answer = i aprreciate your effort .. kind a. I do not wish more details about what he said since i understand that perfectly. If he can add anything nontrivial to it that would be appreciated but i got the impression he gave up ? Anyways : Thank you Robert Israel for your reply :) * shake hands * :) – mick Sep 10 '12 at 14:50
@mick: I was more trying to draw attention to being polite than how to improve the answer. You could just as easily have described what you had done (this can be very important) and then asked, "how should I proceed?" – robjohn Sep 10 '12 at 15:50