How many solutions does the complex equation $$e^z=z$$ have ?   Can we somehow use that this equation implies $z=\ln(z)$ and if this is possible, how do we deal with the infinite many branches of the complex logarithm ?  Is $2$ the number of solutions ?
120661
One of our review problems for my complex analysis final is the following:     Show that $e^z=z$ has infinitely many solutions in $\mathbb{C}$.   I've seen a few solutions online, but none of them have used material that was covered in our introductory complex analysis course. Would anyone be able to provide a simple solution?  I'm trying to think of ways to invoke bounding in some way but I can't seem to come up with anything.
77361
Why does this equation have 6 solutions in $\Bbb{C}$,      $$|z| = z^5$$
82883
I always know that the natural logarithm $\log_e$ is the inverse of the exponential function $e$, but to my surprise when reading elementary functions in complex analysis, I discover that that is not always true. $$\log(e^z)=z+2k\pi i$$  My question here is that, why is the above expression true?
119571
I am investigating solutions to Fermat's equation $$x^n + y^n = z^n$$ with $x,y,z$ in the Gaussian integers, excluding integers and pure imaginaries.  I have found out that there are only trivial solutions for the $n=3$ and $n=4$ cases, e.g. .   I would be grateful if you let me know of the current status or if it is already a theorem.
70109
$$ z = 1+2i \ (complex \ number) \\ z^n = a_n + b_ni \ \ \ (a_n, b_n \in \mathbb{Z}, n \in \mathbb{N}^*) $$ Prove that $ b_{n+2} - 2b_{n+1} + 5b_n = 0$  How can I solve this? Thank you!  EDIT: Or please tell me your ideas.
69403
Let $A$ be a $2 × 2$-matrix with complex entries. The number of $2 × 2$-matrices $A$ with complex entries satisfying the equation $A^3 = A$ is infinite.  Is the above statement true? I know that $0$ and I are two solutions. But are there any more solutions?
27510
I'm trying to find all $z\in \mathbb C$ with $\sin z = 2$ but I'm stuck.  I tried writing $\sin z = {e^{iz} - e^{-iz}\over 2i} = 2$ and then writing $z = x + iy$ but the expression I got was $e^{ix}e^{-y} - e^{-ix}e^{y}=4i$ and I don't see how it's possible to solve this for $x$ and $y$.     Please could someone tell me how to calculate the inverse values of   the complex sine function?   I had also tried letting $w=e^{iz}$. Then I got the quadratic   $$ w^2 - 4i w + 1 = 0$$  and using the formula for quadratics I got  $$ w_{1,2} = {4i \pm i \sqrt{20}\over 2}$$  Taking the logarithm I got  $$ i z_{1,2} = \ln{{4i \pm i \sqrt{20}\over 2}}$$  whereas I should be getting   $$ z = ({\pi \over 2}  + 2\pi k ) - i \ln(2 \pm \sqrt{3})$$  for $k \in \mathbb Z$.
57404
"Find all the possible solutions $(x,y,z) \in$  $\Bbb R$ that satisfies the following equation system".  My teacher gave me this problem and i got some clues in how to solve it. I evaluated $1$ and $-1$ and i got these solutions $(1,-1,1)$ and $(-1,-1,-1)$. I think these are the only ones but i can't prove it.  Am i wrong?, are there more solutions?.   Thanks  $$ \left\{  \begin{array}{c} x+y-z=-1 \\  x^2-y^2+z^2=1\\  -x^3+y^3+z^3=-1 \end{array} \right.  $$
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$z$ is a complex number, by the way.  I've tried a lot of things and always end up with a huge algebraic mess and I've wondered if anyone of you has any idea on how to approach this problem.  One of the most "satisfactory" things I've done so far is to expand both the real and imaginary parts for the equation \vert z + 1/z \vert^2 = a^2. These parts yield the constraints:  $2xy(1+1/(x^2+y^2)^2)=0$ ;   $x^2-y^2+\frac{2(x^2-y^2)}{x^2+y^2}+\frac{(x^2-y^2)}{(x^2+y^2)^2}=a^2 $  I've graphed the equations and supposedly the intersection should be my set of points, but this method is far from analytical.  What can I do? Any hints on how to approach this?
57591
Suppose $f: \overline{\mathbb{D}} \to \mathbb{C}$ is continuous, analytic in $\mathbb{D}$ and satisfies $|f(z)|&lt;1$ for $|z|=1$. Find the number of solutions to the equation $f(z) = z^n$ where $n$ is a positive integer. ($\mathbb{D}$ is the unit disk).  Some of my own attempts: Using the Maximum/Minimum Principle, we can split up the the problem in $2$ parts: Suppose $f$ is constant. It then holds that $z^n = c$ has exactly $n$ zeros because of the fundamental theorem of algebra.  Suppose $f$ is not constant. It holds that $f$ has at least one zero  in $\mathbb{D}$. It of course also holds that $z^n$ has a zero of multiplicity $n$ at the origin. I think I'm supposed to hit with stuff like Rouché's theorem, the Argument Principle or Schwartz lemma, but nothing seems to fit.   Thanks in advance.
73185
Suppose $z$ is a nonzero complex number, so $z=re^{i\theta}$. Show that there are only $n$ distinct complex $n$-th roots, given by $r^{1/n}e^{i(2\pi k+\theta)/n}$ for $0\leq k\leq n-1$.   My proof: Let $z=re^{i\theta}\in\mathbb{C}$ and $a\in\mathbb{C}$ s.t. $a^{n}=z$. Let $a=\rho e^{i\varphi}$. Then, $a^{n}=(\rho e^{i\varphi})^{n}$. Setting $\rho^{n}e^{i\varphi n}=re^{i\theta}$ and rewriting in polar form, we get $$\rho^{n}(\cos(n\varphi)+i\sin(n\varphi))=r(\cos(θ)+i\sin(\theta)).$$ Thus $\rho^{n}=r$ $\rightarrow$ $\rho=r^{1/n}$ and $n\varphi=\theta\rightarrow\varphi=\theta/n$. Thus the roots of $a\in\mathbb{C}$ have the form $$r^{1/n}(\cos(\tfrac{\theta}{n})+i\sin(\tfrac{\theta}{n}))=r^{1/n}(\cos(\tfrac{\theta}{n}+2\pi j)+i\sin(\tfrac{\theta}{n}+2\pi j))$$ for some $j\in\mathbb{Z}$ due to the periodicity of trig functions. Then, $$r^{1/n}(\cos(\tfrac{\theta}{n}+2\pi j)+i\sin(\tfrac{\theta}{n}+2\pi j))=r^{1/n}e^{i\left(2\pi j+\tfrac{\theta}{n}\right)}=r^{1/n}e^{i(\theta+2\pi k)/n}$$ for $0\leq k\leq n-1 ,j=nk\in\mathbb{Z} $.  I feel like there could be a more concise way to put this.
57112
We have the following equation:  $$z^3 = \overline{z} $$  I set z to be $z = a + ib$ and since I know that $ \overline{z} = a - ib$. I was trying to solve it by opening the left side of the equation.  $$ z^3 = (a+ib)^3 \Rightarrow $$ $$ [a^2+b^2+i(ab + ba)](a+ib) \Rightarrow $$ $$ a^3 - b^2a - 2b^2a + i (2a^2b + b^2a - b^3) $$  but this is where I got so far and I'm not sure how continue and if my solution so far is even the right way to solve it.
98209
The equation $X^{n}  + Y^{n}  = Z^{n}$    , where $n   \geq  3$     is a natural number, has no solutions at all where $X,Y,Z$     are integers.   My solution:  False. Because if we let $X=0  ,Y=0$,  then $0^{3}+0^{3}=0^{3}$  . Thus there is a solution for the equation $X^{n}  + Y^{n}  = Z^{n}$,   where $X,Y,Z$   are integers.  Can anyone please give feedback on my answer and state whether it is correct on not.
105231
What solutions does the equation $4x\equiv2\mod10$ have? (Hint, it will have more that one.) What about solutions to an equation $ax\equiv d\mod n$, where $d=\gcd(a,n)$?  This is the question that appeared in our universities last year paper. But it seems very difficult for me like for first part I am able to solve by simplifications I got $x=5\cdot Z +3$ as solution which makes sense and seems fair but for the second one, I have no idea how can I approach that. Can you please help me out with that one. I tried to simplify that by using Euclidean algorithm but still, I didn't get any idea that what actually the solution will be. My final exam is on 2nd of next month and because of covid I am unable to contact in person for any help. So any help from this site will be appreciated. Thank you for looking at this question!
90400
I was trying to solve for $i^i$.  I got distracted and did the following: $e^{i \theta} = \cos(\theta) + i\sin(\theta)$ $e^{i 2\pi} = cos(2\pi) + i\sin(2\pi) = 1$ ||| Take ln() of two sides of equation: $\ln(e^{i2\pi}) = \ln(1)$ $i2\pi = 0$ Clearly this is false. What did I do wrong?
100189
Find the number of solutions to the equation $$2011^x+2012^x+2013^x-2014^x=0$$  The answer seems to be zero, but I have no idea why. Please avoid considering complex solutions and other scary things.
6366
Q1: Could someone provide a proof for this equation (please focus on this question):   $$e^z = \lim \limits_{x \rightarrow \infty} \left( 1 + \frac{z}{x} \right)^x$$  Q2: Is there any corelationn between above equation and the equation below (this would benefit me a lot because i at least know how to proove this one):  $$e = \lim \limits_{x \rightarrow \infty} \left( 1 + \frac{1}{x} \right)^x$$