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$\begingroup$

(There is a question similar to mine, but it talks about f[x] and F[x].)

I've seen some people use F(x) when denoting functions, but what is the difference between using a capital F and a lower case f?

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    No difference in general.... sometimes used as [Antiderivative](https://en.wikipedia.org/wiki/Antiderivative). – 2017-02-22
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    Probably going to depend on context. Capital letters often represent antiderivatives. – 2017-02-22
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    I'm going to say that if you mean it to mean an antiderivative you need to state it is the antiderivative. If you claim f is a function and then refer to another function F, you can't assume it will be understood that F is the antiderivative of f without stating so. But as for notation, you can use any notation you want, just as you can any variable you want. There are conventions based on context but they are not set in stone. – 2017-02-22
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    @fleablood Indeed, there's also the convention that $F$ is the Laplace transform of $f$. – 2017-02-22
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    F could mean all sorts of things. And it could meant anything the writer makes up as well. And that's all fine and makes sense. But there will/should be so contextual indication as to what the writer has in mind. – 2017-02-22

4 Answers 4

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There is no difference. It's a matter of preference. For all you care you could have function $\heartsuit(x)$.

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    How about `\color{red}{\heartsuit}(x)`? – 2017-02-22
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    That's absolutely possible! I would love to see a serious article with a function $\color{red}{\heartsuit}(x)$! – 2017-02-22
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    Set theorists have $\clubsuit$ and $\diamondsuit$... – 2017-02-22
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    Obligatory xkcd: https://www.xkcd.com/55/ – 2017-02-23
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It is sometimes used to denote the antiderivative, but generally there is none. It doesn't matter if you write $f(x)=$ or $F(x)=$ or $\mu(x)=$ or if you use a small picture of a house. You're just choosing what symbol represents the function, it's what comes after the $=$ sign that matters.

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It really depends on the application and field of mathematical study and how the standard notation is set out. I could declare a function f(x) and a function g(x), I could call it whatever i so desire. When working on integrals however, if I integrate f(x), I may choose to call the result F(x). I think the use of capital letters also occurs when working with Fourier Transforms and Laplace Transforms.

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Well, there is absolutely no difference if you are going by notations of functions. But then special meanings are associated with capital and small letters to denote functions and these meanings vary from one field to another. For instance, in statistics, F(x) and f(x) mean two different functions. F(x) represents the cumulative distribution function, or cdf in short, of a random variable as opposed to f(x) which represents the probability density function, or pdf, of the continuous random variable.