 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2015/5/b/i/solution

sinh (x + i y) = \frac{e ^{x} e ^{i y} - e ^{- x} e ^{- i y}}{2} = \frac{e ^{x} ( cos y + i sin y ) - e ^{- x} ( cos y - i sin y )}{2} = \frac{e ^{x} - e ^{- x}}{2} cos y + i \frac{e ^{x} + e ^{- x}}{2} sin y .

Identifying the hyperbolic sine and hyperbolic cosine in the final line gives

sinh (x + i y) = sinh x cos y + i cosh x sin y .

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