1
$\begingroup$

Should you write \begin{equation} \Gamma(z) := \int_{0}^\infty e^{-t}t^{z-1}\,\,dt, \text{ for } \Re(z) > 0, \tag{1} \end{equation} or \begin{equation} \Gamma(z) = \int_{0}^\infty e^{-t}t^{z-1}\,\,dt, \text{ for } \Re(z) > 0?\tag{2} \end{equation}

I thought that $\Gamma(z)$ is defined by the integral $$\int_{0}^\infty e^{-t}t^{z-1}\,\,dt,$$

So $(1)$ is the correct way to introduce the Gamma function in a paper?

Are these differences important when writing a mathematical paper, or subjective and down to personal preference?

$\endgroup$
4
  • 1
    $\begingroup$ It‘s down to personal preference and depends on the context. $\endgroup$
    – Qi Zhu
    Mar 25, 2019 at 14:37
  • 2
    $\begingroup$ That integral is the most common definition of the Gamma function, but not the only one, e.g. some may prefer Euler's infinite product. $\endgroup$ Mar 25, 2019 at 14:50
  • $\begingroup$ @Kezer Okay cool, thanks. $\endgroup$
    – Gurjinder
    Mar 25, 2019 at 14:51
  • $\begingroup$ @RobertIsrael Okay, makes sense, thanks. $\endgroup$
    – Gurjinder
    Mar 25, 2019 at 14:52

1 Answer 1

2
$\begingroup$

It is subjective.

My take is that if you are just reminding the reader of what your notation is, in conjunction with several other well-known facts, the $=$ sign is right. If you are, however, trying to make some subtle point that hangs on what is a definition and what is merely a fact, then the $:=$ is justified.

$\endgroup$
1
  • $\begingroup$ Makes sense, thanks. $\endgroup$
    – Gurjinder
    Mar 25, 2019 at 14:52

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .