Hermite-Lindemann-Weierstrass Theorem/Weaker
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Theorem
Let $a$ be a non-zero algebraic number (possibly complex).
Then:
- $e^a$ is transcendental
where $e$ is Euler's number.
Corollary
Let $a$ be a algebraic number (possibly complex) which is neither $0$ nor $1$.
Then:
- any value of $\ln a$ is transcendental
where $\ln$ denotes complex natural logarithm.
Proof
This follows trivially from Hermite-Lindemann-Weierstrass Theorem by taking $n = 1$.
$\blacksquare$
Source of Name
This entry was named for Charles Hermite, Carl Louis Ferdinand von Lindemann and Karl Theodor Wilhelm Weierstrass.