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Unformatted text preview: sin z = 0 and cos z = 0.
Properties of Trig Functions
1. Adding cos z to i sin z gives Euler's Formula eiz = cos z + i sin z 2. The traditional identities work as usual
sin(z+w) = sinz cosw + cosz sinw
cos(z+w) = cosz cosw - sinz sinw
cos2z + sin2z = 1
3. Real and Imaginary Parts of Sine & Cosine
Some more interesting ones, using (2):
sin(z) = sin(x + iy) = sinx cos(iy) + cosx sin(iy)
sinz = sinx coshy + i cosx sinhy
cosz = cosx coshy - i sinx sinhy
4. dz (sinz) = cos z etc.
Definition 3.5 We also have the hyperbolic sine and cosine,
1 cosh z = 2(ez + e-z)
1 sinh z = 2(ez - e-z)
Note that cosh(iz) = cos z, sinh (iz) = i sin z.
Definition 3.6 A natural logarithm, ln z, of z is defined to be a complex number w such
that ew = z.
1. There are many such numbers w; For example, we know that eiπ = -1. Therefore,
ln(-1) = iπ.
But, eiπ + i2π = -1 as well, therefore,
ln(-1) = iπ + i.(2π)
ln(-1) = iπ +
In general, if
ln z = w,
ln z = w + i(2nπ)
10 2. We calcul...
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