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# C the action of the function gz e z definition 33

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Unformatted text preview: Properties of the Exponential Function 1. For x and y real, eiy = cosy + i sin y and ex is the usual thing. 2. ez ew = ez+w 3. ez /ew = ez-w 4. (ez )w = ezw 5. |eiy| = 1 z z + 2πi 6. Periodicity: e = e dz z 7. Derivative: dz (e ) = e . This follows by either using the Taylor series, or by using the formula ∂u ∂v f'(z) = +i ∂x ∂x Examples 3.2 (a) We compute e3+2i, and e3+ai for varying a. (b) The geometric action of the exponential function: it transforms the complex plane. Vertical lines go into circles. The vertical line with x-coordinate a is mapped onto the circle with radius ea. Thus the whole plane is mapped onto the punctured plane. (c) The action of the function g(z) = e-z. Definition 3.3 Define the trigonometric sine and cosine functions by 1 cos z = 2(eiz + e-iz) 9 1 sin z = 2i(eiz - e-iz) (Reason for this: check it with z real.) Similarly, we define sin z tan z = , cos z etc. Examples 3.4 (A) We compute the sine and cosine of π/3 + 4i (B) Determine all values of z for which...
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