How can $i^i = e^{-\pi/2}$ !!
I was asked a homework question: find $i^i$. The solution provided was as follows: Let $A = i^i$. $\log A = i \log i$. Now, $\log i = \log e^{i\pi/2} = \f...
Apr 21, 2026 1 VIEWS
Topic: General
5104 Articles 
Solving $\cos z = i$ for $z$
Solve $\cos z = i$ for $z$. What I have tried: $$\cos z = i$$ $$\frac{e^{-zi}+e^{zi}}{2}=i$$ $$e^{-zi}+e^{zi}=2i=2e^{\frac\pi 2 + 2\pi k},\quad k\in \Bbb ...
Apr 21, 2026 1 VIEWS
a probability question using percentages
This question is confusing me as I am not used to seeing percentages in a possibility question. in a large insurance agency - 60% of the customers have au...
Apr 21, 2026 1 VIEWS
proof of the Meyers-Serrin Theorem in Evans's PDE book
The following is the Meyers-Serrin theorem and its proof in Evans's Partial Differential Equations: Could anyone explain where (for which $x\in U$) i...
Apr 21, 2026 1 VIEWS
Changing order of integration in cylindrical coordinates
I'm having a problem in changing order of integration in triple integration, in cylindrical coordinates. I would be grateful for a little help.The qu...
Apr 21, 2026 1 VIEWS
Convergence in Lp implies convergence in Lp norms finite
I have the following theorem which I believe is true: Suppose we have the measure space $(\mathscr{X},\mathcal{A}, \mu)$ and $p\in [1,\infty)$. Let $\{f_n...
Apr 21, 2026 1 VIEWS
Pattern for all the binary chains divisible by 5
For instance, $x = 101$ is divisible by $5$ because it is the integer 5. Same thing for $x=1111$ is also divisible by 5 as it is the integer 15. However, ...
Apr 21, 2026 1 VIEWS
Prove radius of fourth circle is equal to semi-perimeter of the right triangle formed by three circles
Question: There are 3 circles that touch. A right triangle forms the center of the circles. Find the radius of the fourth circle given that it is circumsc...
Apr 21, 2026 2 VIEWS
Calculating Adjacency Matrix
I'm having trouble understanding the concept, I know it is pretty simple but could someone help me out. Assume that I have the following: $V = \begin...
Apr 21, 2026 1 VIEWS
Is there more than one possible reduced row echelon form of a matrix?
I'm trying to find the rref(A) where A = \begin{bmatrix}2&1&4&-4&11\\1&2&3&1&8\\1&1&2&-1&6\end{bmatri...
Apr 21, 2026 1 VIEWS
Quotient algebra of tensor algebra
There is the definition of exterior algebra over $R$-module $M$ as quotient algebra of tensor algebra: $$\Lambda(M):= \frac{T(M)}{I}$$ Where $I$ is the id...
Apr 21, 2026 1 VIEWS
Name of non-complete graph with certain properties
Is there a name for a graph with the following criteria? not a complete graph has at least n vertices (let's say n > 3 to exclude trivial cases) w...
Apr 21, 2026 1 VIEWS