Intereting Posts

Non-Probabilistic Argument for Divergence of the Simple Random Walk
Necessity/Advantage of LU Decomposition over Gaussian Elimination
Time required to reach the goal when an object will be slowing down incrementally based on distance travelled?
Does the Rational Root Theorem ever guarantee that a polynomial is irreducible?
Finding the general solution of the differential equation $\,\,y''+y=f(x)$
Math Intuition and Natural Motivation Behind t-Student Distribution
Deducing a $\cos (kx)$ summation from the $e^{ikx}$ summation
Limit of this recursive sequence and convergence
Determinant of a block lower triangular matrix
Compute $\int_0^\infty\frac{\cos(xt)}{1+t^2}dt$
$ \lim_{x\rightarrow 0^{+}}\frac{\sin ^{2}x\tan x-x^{3}}{x^{7}}=\frac{1}{15} $
Something Isn't Right With My Parking
Introductory Linear Algebra Book Recommendation
$f^2+2f+1$ is a polynomial implies that $f$ is a polynomial
Basic Taylor expansion question

Is $\large i^{i^i}$ real ? How to find it?

Thank You!

- Factoring algebraic expressions of three variables
- What are better approximations to $\pi$ as algebraic though irrational number?
- How prove this $\frac{a}{(bc-a^2)^2}+\frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2}=0$
- Calculus book recommendations (for complete beginner)
- Why not write the solutions of a cubic this way?
- How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

- Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?
- Does $\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}$ converge conditionally?
- Lesser-known integration tricks
- Nested Square Roots
- The notation for partial derivatives
- Gaussian-Like integral
- Basic Taylor expansion question
- Integral of $\frac{1}{x^2+x+1}$ and$\frac{1}{x^2-x+1}$
- What is wrong with this funny proof that 2 = 4 using infinite exponentiation?
- Improper integral of $\frac{x}{e^{x}+1}$

$i^i=e^{i\log i}$

Now on principal branch,using $i=e^{i\pi/2}\implies \log i=i\pi/2$ gives $i^i=e^{-\pi/2}$

Therefore, $i^{i^i}=i^{e^{-\pi/2}}=e^{e^{-\pi/2}\log i}=e^{i(\pi e^{-\pi/2})/2}=\cos\left(\pi \frac{e^{-\pi/2}}{2}\right)+i\sin\left(\pi \frac{e^{-\pi/2}}{2}\right)$

and hence its imaginary part is $\neq 0$ as $ \frac{e^{-\pi/2}}{2}$ is not an integer.

Complex powers may have more than one value. In our case

$$ i^i=e^{i \log i}=\exp \left(i \left(\ln(1)+i \frac{\pi}{2}+2\pi k i\right)\right)=e^{-\frac{\pi}{2}+2\pi k} $$ where $k$ is an integer. Thus $$i^{i^i}=e^{i^i \log(i)}=\exp\left(e^{-\frac{\pi}{2}+2\pi k}\cdot\left(i \frac{\pi}{2}+2\pi l i\right)\right), $$

which is $e$ to an imaginary power. It is therefore a point on the unit circle, but it can never be chosen real.

Wolfram Alpha gives the answer to be:

$0.94715899… + 0.320764449… i$.

Therefore it is not a real number, as the answer has an imaginary component to it.

We have $i^i=(e^{i\pi /2})^i=e^{-\pi /2}$. Then, $$i^{i^i}=i^{e^{-\pi /2}}=(e^{-i\pi /2})^{e^{-\pi /2}}=e^{-i\pi e^{-\pi /2}/2}=\cos (\pi e^{-\pi /2}/2)-i\sin(\pi e^{-\pi /2}/2) $$ Now $\sin(\pi e^{-\pi /2}/2)$ is non-zero, since $e^{-\pi /2}/2$ is not an integer.

First find all the values of $i^i$. Then, if $\beta$ is one of those values, a possible value of $i^{i^i}$ is $(e^{i\pi/2})^{\beta}$. Or $(e^{-3i\pi/2})^{\beta}$.

- Recurrent problem about polynomials
- Why is the range of inverse trigonometric functions defined in this way?
- integral of complex conjugate times the differential is purely imaginary
- Prove that $2^{2^{\sqrt3}}>10$
- Double Integral of xy
- Simple proof that equilateral triangles have maximum area
- A stick of unit length is cut into three pieces of lengths $X, Y, Z$ according to its length in two sequence of cuts. Find Cov(X,Y).
- Are there straightforward methods to tell which function has fastest asymptotic growth without a calculator?
- Is $2^\infty$ uncountable and is cardinality a continuous function?
- Number-theoretic asymptotic looks false but is true?
- Multiplication of Set Discrete math
- $f\cap f\supsetneq f$ – Where does the string of equivalences fail ?
- One divided by Infinity?
- First-order trigonometric differential equation
- Why does the sign have to be flipped in this inequality?