Is the equation solvable in the complex field? Except for the class of x x = 5, including transcendental equation such as SiNx = 6, and please explain the reason (I mean that there are complex numbers satisfying the equation in theory, which need not be solved)

If the restriction is an algebraic equation, then it is solvable. This is Gauss's basic theorem of algebra. If it is a transcendental equation, it is not necessary. Picard has a theorem that an entire function can take any value except one. For example, e ^ x = 0 has no solution. More complex functions are not necessarily

Solving the equation in complex field: sinhz + coshz = 0

sinhz=(e^z-e^(-z))/2,coshz=(e^z+e^(-z))/2
sinhz+coshz=e^z
Let z = x + iy, x, y be real numbers, then e ^ z = e ^ (x + iy) = (e ^ x) * e ^ (iy) = (e ^ x) * (cosy + isiny)
If e ^ z = 0, cosy = 0, siny = 0
There is no solution

It is proved that the orthogonal matrix of the upper triangle must be a diagonal matrix, and the elements on the main diagonal are positive 1 or negative 1

Let a = [A1, A2,..., an] be the orthogonal matrix of the upper triangle
a1=(a11,0,...,0)^T,a2=(a12,a22,0,...,0)^T,...,an=(a1n,a2n,...,ann)(akk≠0,k=1,2,...,n)
From A1 ^ t * AK = 0 (K ≠ 1), we get: a11 * a1k = 0, that is, a1k = 0 (k = 2,3,..., n)
Similarly, AIJ = 0 (I)

Is the equation solvable in the complex field? Except for the class of x x = 5, including transcendental equation such as SiNx = 6, and please explain the reason (I mean that there are complex numbers satisfying the equation in theory, which need not be solved)