Intereting Posts

Fourier Series on a 2-Torus
Convergence of sequence given by $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$
Evaluating definite integrals
Implicit differentiation
$g'(x) = \frac{1}{x}$ for all $x > 0$ and $g(1) = 0$. Prove that $g(xy) = g(x) + g(y)$ for all $x, y > 0$.
Why is this congruence true?
If ${n \choose 3} + {n+3-1 \choose 3} = (n)_3$, compute $n$.
Sum of squares diophantine equation
Are axioms assumed to be true in a formal system?
Calculating the Modular Multiplicative Inverse without all those strange looking symbols
Evaluate $\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$
Quadratic Formula in Complex Variables
How does Mathematica solve $f(x)\equiv 0\pmod p$?
Find a polynomial of degree > 0 in $\mathbb Z_4$ that is a unit.
Find all solutions to $x^9 \equiv 25$(mod 29)

Joseph Bak and Donald Newman’s complex analysis book (p.236) has a proof that the equation $e^z-z=0$ has infinitely many complex solutions:

I’m curious if there are any particularly elegant ways to see this, other than that given in the text.

- Presheaf which is not a sheaf — holomorphic functions which admit a holomorphic square root
- Complex Mean Value Theorem: Counterexamples
- Complex towers: $i^{i^{i^{…}}}$
- If $f \in \operatorname{Hol}(D)$, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$
- Use Picard's Theorem to prove infinite zeros for $\exp(z)+Q(z)$
- Understanding branch cuts for functions with multiple branch points

- Infinite Series $\sum\limits_{n=1}^{\infty}\frac{1}{\prod\limits_{k=1}^{m}(n+k)}$
- the real part of a holomorphic function on C \ {0, 1}
- Why is continuous differentiability required?
- What would be the value of $\sum\limits_{n=0}^\infty \frac{1}{an^2+bn+c}$
- Conformal map between annulii
- Lagrange inversion theorem application
- Proving two entire functions are constant.
- Proving that if $|W(-\ln z)| < 1$ then $z^{z^{z^{z^…}}}$ is convergent
- How to prove Lagrange trigonometric identity
- Evaluate a definite integral involving Airy functions

An elementary proof: Let $z = x +y i$ then $|e^z| = |z|$ precisely if $e^{2x} = x^2 + y^2$. If $x \geq 0$ then $e^{2x} – x^2 > 0$ so $y = (e^{2x} – x^2)^{1/2}$ is a positive solution of this equation. This means that for all $x \geq 0$ there is such a $y \geq 0$ such that $|e^z| = |z|$. The argument of $z$ is in $[0, \pi/2]$ since $x, y \geq 0$. The argument of $e^z$ is $y$. Since $y \to \infty$ when $x \to \infty$ there are infinitely many such $z$ for which both $|e^z|=|z|$ and $\arg(e^z) \equiv \arg (z) \pmod {2\pi}$. These $z$ are therefore roots of $e^z-z$.

If you use the fairly deep result of Picard about essential singularities then you can prove this as follows: $f(z) = e^z-z$ has an essential singularity at infinity. Therefore $f$ attains all values infinitely many times with at most one exception (that is, at most a single value could be attained only finitely many times). This exception could still be $0$. However, $f$ also satisfies $f(z + 2\pi i) = f(z) – 2\pi i$. Now $f$ attains at least one value in $\{0, 2\pi i\}$ infinitely many times. In both cases it follows that $f$ must have infinitely many zeroes.

One way to see this is to realize that $f(z)=e^z$ has $\{z\in\mathbb{C}:0\leq\mathrm{Im}(z)<2\pi\}$ as a fundamental region and has period $2\pi i$. That fundamental region is mapped onto the plane (excluding $0$), as is every shift of the region by integer multiples of $2\pi i$. From there, it isn’t difficult to show that there must be infinitely many $z\in\mathbb{C}$ for which $f(z)=z$.

It remains only to show (as pointed out below by Harald) that in each such shift of the region there is at least one solution–that is, at least one zero of the function $g(z)=e^z-z$. Harald’s suggestion of applying the argument principle (see http://en.wikipedia.org/wiki/Argument_principle if needed) is a good one. Noting that the function is entire (so no poles), you really need only show that $\oint_C\frac{e^z}{e^z-z}dz$ is non-$0$ (where $C$ is the contour he suggests) for sufficiently large $M$.

- How to tell if some power of my integer matrix is the identity?
- For every continuous function $f:\to$ there exists $y\in $ such that $f(y)=y$
- Lebesgue measurability of a set
- The sum of the elements in a field of at least three elements is 0
- Can I apply the Girsanov theorem to an Ornstein-Uhlenbeck process?
- why is a simple ring not semisimple?
- Cayley-Hamilton theorem on square matrices
- Find the density of the sum of two uniform random variables
- Characterize the integers $a,b$ satisfying: $ab-1|a^2+b^2$
- Mean and variance of the order statistics of a discrete uniform sample without replacement
- Step forward, turn left, step forward, turn left … where do you end up?
- Solving a ODE from a diffeomorphism numerically
- Integral involving the hyperbolic tangent
- Existence of irreducible polynomials over finite field
- Can equinumerosity by defined in monadic second-order logic?