# Is the Gamma function defined by $\int_{0}^\infty e^{-t}t^{z-1}\,\,dt?$

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

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?

• It‘s down to personal preference and depends on the context. Mar 25, 2019 at 14:37
• 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. Mar 25, 2019 at 14:50
• @Kezer Okay cool, thanks. Mar 25, 2019 at 14:51
• @RobertIsrael Okay, makes sense, thanks. Mar 25, 2019 at 14:52

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.