Does this trigonometric equation: z 2 </mfrac> csc &#x2061;<!-- ⁡ -->

Ellie Benjamin

Ellie Benjamin

Answered question

2022-07-16

Does this trigonometric equation: z 2 csc ( z 2 ) = 1, have solutions other than zero?

Answer & Explanation

Gornil2

Gornil2

Beginner2022-07-17Added 20 answers

If you call z=x+iy, then solving the equation
z c s c z = 1
Is equivalent to solve the pair of equations
sin x cosh y = x and cos x sinh y = y
This problem can be numerically solved using Newton's method. Note that this system can be also written as
tan x x = tanh y y = α
That allows you to see that there are solutions different from z=0, indeed if you fix α < 0, the problem is reduced to finding roots of tanh y = α y and tan x = α x, which exist, but do not have a closed algebraic form.

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