Hyperbolic function

[Agent Smith]

Which is correct
1. $\displaystyle \cosh x = \frac{e^x + e^{-x}}{2}$

or

2. $\displaystyle \cosh x := \frac{e^x + e^{-x}}{2}$

Thank you

 

[khansaheb]

Which is correct
1. $\displaystyle \cosh x = \frac{e^x + e^{-x}}{2}$

or

2. $\displaystyle \cosh x := \frac{e^x + e^{-x}}{2}$

Thank you

What is/are the difference/s between (1) and (2) - as described in the op?

 

[Agent Smith]

What is/are the difference/s between (1) and (2) - as described in the op?

All I know is $\displaystyle :=$ means "is defined as" and $\displaystyle =$ means "is equal to".

 

[Dr.Peterson]

All I know is $\displaystyle :=$ means "is defined as" and $\displaystyle =$ means "is equal to".

So, you're asking whether that formula is the definition, or a result of some other definition?

That depends on what you use as the definition! Did you try looking it up?

Hyperbolic functions - Wikipedia

en.wikipedia.org

You'll see that you can start with various definitions, that formula being one of them. So your question is moot. The answer is, it can be, or not.

 

[Agent Smith]

So, you're asking whether that formula is the definition, or a result of some other definition?

That depends on what you use as the definition! Did you try looking it up?

Hyperbolic functions - Wikipedia

en.wikipedia.org

You'll see that you can start with various definitions, that formula being one of them. So your question is moot. The answer is, it can be, or not.

In natural language we use ":=" to (re)define a (new/old) word/symbol. E.g. from now on "xt" means "the wolf without a tail" i.e. $\displaystyle \text{xt}:= \text{wolf without a tail}$. Then you know that when I write "xt ate the old man", I mean the wolf without a tail ate the old man. My hunch is that it's a part of a given $\displaystyle \mathbb{D}$ (domain of discourse), in which you want to use a special word/symbol to mean something specific for as long as we're within $\displaystyle \mathbb{D}$.

$\displaystyle \cosh$ is just a group of symbols and is being defined as $\displaystyle \frac{e^x + e^{-x}}{2}$