how to finish the input for Euler's formula expression

You don't have to define the formula as Mathcad and also Prime are aware of it.

And what do you mean by  "finish the input" ?

As you see in the picture, the imaginary unit must be input as "1i" (without the quotes and without a multiplication in-between).

Unfortunately in Prime the imaginary unit is also shown that way. That's nicer looking in old Mathcad (here MC15 in the pic)

As Werner says, Mathcad is already aware of the relationship - Mathcad numbers are complex by default and many functions are complex-aware.

However, if you wanted to define your own Euler equation, then you can do so as follows:

E(x) := cos(x) + 1i*sin(x)

where x is real.

Euler deserves a capital, and it avoids confusion with the built in exponent function.

Stuart

Appreciate it a lot, it's so clear to my confusion.