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#### Tranquillity

##### Member

- Feb 22, 2012

- 36

My approach:

cos(z)= [ e^(iz) + e^(-iz) ] / 2 = 2i

Rearranging and setting e^(iz) = w

we get a quadratic w^2 - 4iw + 1 = 0

The quadratic yields two solutions:

w=e^(iz) = i(2 + sqrt(5))

or e^(iz) = i(2-sqrt(5))

And now my problem is here.

In the lectures we are only using the principal logarithm ie

"Let D0 be C with the origin and the negative real axis removed. Deﬁne, forz in D0,

w = Logz = ln|z| + iArgz.

Here Argz ∈ (−pi, pi) is the principal argument"

Moreover, we have been given that

"It is not always true that Log (exp(z)) = z. e.g. z = 2πi gives

exp(z) = 1, Log (exp(z)) = Log 1 = 0."

So actually with what I was taught during lectures I cannot just take logarithms of both sides.

How should I solve the equation?

Thank you for all the help!