Intereting Posts

Normal subgroups and cosets
Show that exist a unique expression for $ A $ of the form $ A = (A_1 + A_2) + i (B_1 + B_2) $
Why does $1/x$ diverge?
Is the composition of $n$ convex functions itself a convex function?
Induced maps in homology are injective
How to prove $\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1$
How to calculate the inverse of a point with respect to a circle?
Finding an equation of circle which passes through three points
Maximal submodule in a finitely generated module over a ring
Induction proof of $n^{(n+1) }> n(n+1)^{(n-1)}$
Show that $S_4/V$ is isomorphic to $S_3 $, where $V$ is the Klein Four Group.
Inverse of the Joukowski map $\phi(z) = z + \frac{1}{z}$
What is the fastest numeric method for determinant calculation?
Prove that $(a-b) \mid (a^n-b^n)$
Definition of convolution?

Are exponential & trigonometric functions the only non-trivial solutions to $F'(x)=F(x+a)$?

$F(x)=0$ would be the trivial solution. Then, for $a=0$ (or $a=2\pi i$), we have $F(x)=e^x$, and for $a=\dfrac\pi2$ there are $F(x)=\sin x$ and $F(x)=\cos x$. But the three are connected by *Euler’s formula* $e^{ix}=\cos x$ $+i\sin x$. Indeed, on a more general note, letting $F(x)=e^{\lambda x}$, we have $\lambda=\dfrac{W(-a)}{-a}$ where *W* is the *Lambert W function*. My question would be if these are the *only* ones, due to the *special* properties of the number *e* and the *exponential function*, or if there aren’t by any chance *more*, which do *not* belong in the same family or category as these, i.e., which are not *exponential* or *trigonometric* in nature ? Thank you.

- Gaussian-like integral : $\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$
- Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?
- Show that a continuous function has a fixed point
- Behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$
- Prob. 9, Chap. 6, in Baby Rudin: Integration by parts for improper integrals
- On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)

- Can a non-zero vector have zero image under every linear functional?
- Proof for Dirichlet Function and discontinuous
- Why isn't an odd improper integral equal to zero
- If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?
- Two different expansions of $\frac{z}{1-z}$
- Residue Formula in complex analysis
- Convergence of $\sum_{n=1}^{\infty }\frac{a_{n}}{1+na_{n}}$?
- Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?
- A problem in the space $C$
- Prob. 2, Chap. 6, in Baby Rudin: If $f\geq 0$ and continuous on $$ with $\int_a^bf(x)\ \mathrm{d}x=0$, then $f=0$

Look at these MO entries:

https://mathoverflow.net/questions/114875/on-equation-fz1-fz-fz/114878#114878 ,

and

https://mathoverflow.net/questions/156312/solve-fx-int-x-1×1-ft-mathrmdt/156315#156315 .

They contain the answer to your question.

EDIT. To put it shortly, the answer is: “yes” and “no”.

In exactly the same sense as the answer to a simpler question:

“Is every periodic function an

exponential/trigonometric sum”? “Yes” for a physicist, and “no” for a mathematician.

But every periodic function is a limit of exp/trig sums.

- Surjective Function from a Cantor Set
- Showing that a $3^n$ digit number whose digits are all equal is divisible by $3^n$
- A formula for the roots of a solvable polynomial
- zero set of an analytic functio of several complex variables
- Is there a way to define the “size” of an infinite set that takes into account “intuitive” differences between sets?
- How many shapes can one make with $n$ square shaped blocks?
- Alternative definition of $\|f\|_{\infty}$ as the smallest of all numbers of the form $\sup\{|g(x)| : x \in X \}$, where $f = g$ almost everywhere
- Why is a genus 1 curve smooth and is it still true for a non-zero genus one in general?
- Calculating equidistant points around an ellipse arc
- In Triangle, $\sin\frac A2\!+\!\sin\frac B2\!+\!\sin\frac C2\!-\!1\!=\!4\sin\frac{\pi -A}4\sin\frac{\pi -B}4\sin\frac{\pi-C}4$
- Prove that $f(z)=\frac{1}{2\pi}\int_0^{2\pi} f(Re^{i\phi})Re(\frac{Re^{i\phi}+z}{Re^{i\phi}-z}) d\phi$
- Method of Exhaustion applied to Parabolic Segment in Apostol's Calculus
- Radius of a cyclic quadrilateral given diagonals
- Find the eigenvalues of a matrix with ones in the diagonal, and all the other elements equal
- For a field $K$, is there a way to prove that $K$ is a PID without mentioning Euclidean domain?