How can I solve cosh(z)=2 for z∈C? My steps have only led to values of z which are not imaginary, those being ln⁡(2+3) and ln⁡(2−3). How do I find solutions to the equations that are imaginary?

Using the fact that e2πi=1, if ex=2+3, then ex+2πi=ex+4πi=ex+2nπi….=2+3.A similar argument for ex=2−3Therefore, the solutions are ln⁡(2±3)+2nπi

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Samara Richard

2022-03-26

How can I solve

cosh (z) = 2

for

z \in C

? My steps have only led to values of z which are not imaginary, those being

\ln (2 + \sqrt{3})

and

\ln (2 - \sqrt{3})

. How do I find solutions to the equations that are imaginary?

Kingston Lowery

Beginner2022-03-27Added 10 answers

Using the fact that

e^{2 π i} = 1

, if

e^{x} = 2 + \sqrt{3}

, then

e^{x + 2 π i} = e^{x + 4 π i} = e^{x + 2 n π i} \dots . = 2 + \sqrt{3}

.
A similar argument for

e^{x} = 2 - \sqrt{3}

Therefore, the solutions are

\ln (2 \pm \sqrt{3}) + 2 n π i

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