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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 + 2πi
6. Periodicity: e = e
7. Derivative: dz (e ) = e .
This follows by either using the Taylor series, or by using the formula
(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
tan z =
(A) We compute the sine and cosine of π/3 + 4i
(B) Determine all values of z for which...
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