Intereting Posts

Pointwise convergence does not imply $f_n(x_n)$ converges to $f(x)$
Why square matrix with zero determinant have non trivial solution
Geometry question regarding existence of a quadrilateral
Sum of two squares $n = a^2 + b^2$
How to geometrically prove the focal property of ellipse?
Calculate volume of intersection of 2 spheres
Help to understand proof to show that a function is uniformly continous in a certain interval (Spivak)
Proof that sum of first $n$ cubes is always a perfect square
On Diophantine equations of the form $x^4-n^2y^4=z^2$
Does the following have a solution for f(x,y)?
Find the Mean for Non-Negative Integer-Valued Random Variable
Applications of Principal Bundle Construction: Vague Question
Difference of order statistics in a sample of uniform random variables
$a_0 = 5/2$. $a_k = (a_{k-1})^{2} – 2$ for $k\geq1. \prod_{k=0}^{\infty}{\left(1-1/a_k\right)}.$
How to show that $\lim_{n\to \infty} f_n(x) = 0$.

Here’s my (obviously flawed) proof that $1=e^{-2 \pi}$:

$$

1^i=1\\

e^{2 \pi i} = 1\\

\left(e^{2\pi i}\right)^i = 1^i\\

e^{-2 \pi} = 1

$$

What’s the issue? I understand that exponentiation is not injective (and thus $-1 \neq 1$ even though $(-1)^2 = 1^2$), but I don’t think that’s an issue here: I’m only raising things to the power of $i$, which I don’t think is multi-valued.

- Gaussian integral with a shift in the complex plane
- Properties of the Mandelbrot set, accessible without knowledge of topology?
- Maximum of $|(z-a_1)\cdots(z-a_n)|$ on the unit circle
- Convergence of Taylor Series
- Existence of continuous and analytic function
- How does one define the complex distribution $1/z$?

- Zero sum of roots of unity decomposition
- Show that $f$ is a polynomial of degree $\le n$
- Why do we categorize all other (iso.) singularities as “essential”?
- Determine a holomorphic function by means of its values on $ \mathbb{N} $
- Number of roots of $x^a-1=0$ with $a \in \mathbb{C}$
- Analytic functions in a punctured disk
- Fourier transform of $\frac{1}{1+x^2}$
- Find all complex numbers satisfying the equation.
- Prove the open mapping theorem by using maximum modulus principle
- Riemann zeta function and the volume of the unit $n$-ball

In complex numbers, either $x^y$ is a multivalued function, or you have to give up the notion that $(x^y)^z = x^{yz}$.

If you allow $x^y$ to be multi-valued, then one of the values for $1^i$ is $e^{-2\pi}$.

If $x^y$ is not multivalued, then you have to pick a single value for $\log 1$ to define $1^y$. We usually pick $\log 1 = 0$, for some reason. ðŸ™‚

The multivalued nature makes sense when you consider that $\sqrt{1}=1^{1/2}$ can be thought of as having two values, $-1$ and $1$. In general, though, when $y$ is irrational, you get $1^y$ (or more generally, $x^y$) can take infinitely many values.

The only time $x^y$ is naturally single-valued is when $y$ is an integer.

The law of exponents $x^{ab} = (x^a)^b$ does not hold in general.

Raising numbers to complex powers is multivalued.

For example, $1^i = e^{\log(1^i)} = e^{i (\ln(1)+2\pi k i)} = e^{i(2\pi k i)} = e^{-2\pi k}$ for any integer $k$. Choosing $k=-1$ matches your left hand side, $k=0$ matches the right.

- The Krull dimension of a module
- Prove that function is homeomorphism.
- The number of summands $\phi(n)$
- show that $19-5\sqrt{2}-8\sqrt{4}$ is a unit in $\mathbb{Z}{2}]$
- Why is $\tan((1/2)\pi)$ undefined?
- Cofinality and its Consequences
- How should I calculate the $n$th derivative of $f(x)=x^x$?
- Equivalence to the prime number theorem
- Why do units (from physics) behave like numbers?
- strong induction postage question
- Curvature of Ellipse
- Equality of integrals
- a question about fixed-point-free automorphism
- Given a group $ G $, how many topological/Lie group structures does $ G $ have?
- Is there a connection between the diagonalization of a matrix $A$ and that of the product $DA$ with a diagonal matrix $D$?