MHB Is e^x a transcendental function?

[highmath]

Why The function e^z is transcendental over C(z)?

 

[HOI]

What the definition of "transcendental function"?

 

[highmath]

A function that its result, f(x) is not can be found by addition, subtraction, division, multiplication, roots and powers operation.
Still, I don't understand.

 

[HOI]

Well, given a value of x, say, $x= \pi$, how would you find f(x)? What is $f(\pi)= e^\pi$? (I can think of two different ways but neither would give an exact numerical value. I wouldn't expect them to because this is an irrational number.)

 

[highmath]

O.K.
In my words, because e with any power of any number is transcedental function...
Right?

 

[HOI]

Well, that's just restating that e^x is a transcendental function isn't it? The "two ways" or evaluating, say, e^\pi are
(1) Use a calculator! The calculator typically uses the "CORDIC" method (https://en.wikipedia.org/wiki/CORDIC)
(2) Evaluate the Taylor's series expansion for e^\pi. That is 1+ \pi+ \frac{\pi^2}{2}+ \frac{\pi^3}{3!}+ \cdot\cdot\cdot for some finite number of terms. Strictly speaking, that is a polynomial calculation. But e^x is NOT a polynomial because the exact value requires the infinite series.