Is e^x a transcendental function?

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Discussion Overview

The discussion centers around whether the function e^x is considered a transcendental function, exploring definitions, evaluations, and implications of transcendentality in the context of complex numbers and irrational values.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions why the function e^z is transcendental over C(z).
  • Another participant seeks clarification on the definition of "transcendental function."
  • A participant proposes that a transcendental function is one whose result cannot be derived from basic arithmetic operations or roots.
  • There is a discussion on evaluating e^π, with one participant noting that exact numerical values cannot be obtained for irrational numbers.
  • Another participant asserts that e raised to any power is a transcendental function, seeking confirmation of this understanding.
  • A later reply emphasizes that evaluating e^π can be done through methods like using a calculator or Taylor's series expansion, but notes that the exact value requires an infinite series, distinguishing it from polynomial calculations.

Areas of Agreement / Disagreement

Participants express differing views on the definition and implications of transcendental functions, with no consensus reached on the nature of e^x as a transcendental function.

Contextual Notes

Participants have not fully resolved the definitions and implications of transcendental functions, and there are assumptions regarding the evaluation methods that remain unaddressed.

highmath
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Why The function e^z is transcendental over C(z)?
 
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What the definition of "transcendental function"?
 
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.
 
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.)
 
O.K.
In my words, because e with any power of any number is transcedental function...
Right?
 
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.
 

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