Zeroes of linear combination of sines [closed]

Question

Let us consider $$f(z):=\sum\limits_{j=1}^{j=n}a_j\sin(\lambda_jz) $$ where all $a_j$ and $\lambda_j$ (of course, $\lambda_j$ are distinct) are real numbers and $n \in \mathbb{Z},\, n \ge 3$. The question is: are all the zeroes of $f(z)$ real?

Answer 1

Note that $\sinh$ is a strictly increasing positive unbounded function on $(0,\infty)$ so for say $n \ge 3$, the equation $\sinh 1+ \sinh 2+...\sinh (n-1) =\sinh x$ has a unique positive solution $x_n> n-1$.

Using that $\sin {iy}=i\sinh y$ for real $y$, the above means that $i$ is a root of the equation $\sin z+\sin 2z +..\sin {(n-1)z}-\sin x_nz=0$

For $n=2$ we can adapt this and use that $2\sinh 1=\sinh x_2$ for a unique $x_2>1$ and get that $i$ is a root of $2\sin z-\sin x_2z$

Answer 2

Proof of the existence of a complex root in the general case, expanding Oleg's remark:
Think of the $n$ complex numbers $\sin(\lambda_j z)$ as vectors in the 2D plane; for $n\geq 3$ and fixed $z$ add the first $n-2$ of these vectors with coefficients $a_j\neq 0$ such that the sum is neither a real multiple of $\sin\lambda_n z$ nor of $\sin\lambda_{n-1}z$; then adjust $a_n\neq 0$ and $a_{n-1}\neq 0$ to cancel the full sum.

---

an example for $n=10$ (requested by the OP): $$\sum_{n=1}^8 \sin nz +\tfrac{1}{2}\sin 9z+\sin 10z=0$$ for $z=0.6142+0.0627\,i$.

To demonstrate that this zero is not a numerical artefact, here is a plot in the complex plane of contours where the real part of the sum vanishes (orange) and of contours where the imaginary part vanishes (blue); the contours intersect, demonstrating the existence of a complex root.

_{The Mathematica code (contour plot in the complex plane of the roots of the real and imaginary parts of sum of sines) :
ContourPlot[{Cosh[y] Sin[x] + Cosh[2 y] Sin[2 x] + Cosh[3 y] Sin[3 x] + Cosh[4 y] Sin[4 x] + Cosh[5 y] Sin[5 x] + Cosh[6 y] Sin[6 x] + Cosh[7 y] Sin[7 x] + Cosh[8 y] Sin[8 x] + 1/2 Cosh[9 y] Sin[9 x] + Cosh[10 y] Sin[10 x] == 0, Cos[x] Sinh[y] + Cos[2 x] Sinh[2 y] + Cos[3 x] Sinh[3 y] + Cos[4 x] Sinh[4 y] + Cos[5 x] Sinh[5 y] + Cos[6 x] Sinh[6 y] + Cos[7 x] Sinh[7 y] + Cos[8 x] Sinh[8 y] + 1/2 Cos[9 x] Sinh[9 y] + Cos[10 x] Sinh[10 y] == 0}, {x, 0.614, 0.6145}, {y, 0.06265, 0.06275}, WorkingPrecision -> 30]}

---

the OP also requests an example for arbitrary $n$, let me take $z=\pi +i$ and $\lambda_k=k$, and all coefficients $a_k$ equal to unity except $a_n=a$, $$S_{n}=\sum_{k=1}^{n-1}\sin k(\pi +i)+a\sin[n(\pi+i)]=$$ $$\qquad\qquad=-\frac{i}{2 \cosh\left(\frac{1}{2}\right)} \left[(-1)^{n} \sinh \left(n-\tfrac{1}{2}\right)+\sinh \left(\tfrac{1}{2}\right)\right]+a(-1)^ni\sinh n,$$ which vanishes for a nonzero real $a$.