Math144Notes

# Logarithms definition 36 a natural logarithm ln z of

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 &amp; Cosine Some more interesting ones, using (2): sin(z) = sin(x + iy) = sinx cos(iy) + cosx sin(iy) sinz = sinx coshy + i cosx sinhy and similarly cosz = cosx coshy - i sinx sinhy d 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. Logarithms Definition 3.6 A natural logarithm, ln z, of z is defined to be a complex number w such that ew = z. Notes 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) Similarly, ln(-1) = iπ + In general, if ln z = w, then ln z = w + i(2nπ) 10 2. We calcul...
View Full Document

Ask a homework question - tutors are online