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Complex Variables : Stereographic Projections, Hyperbolic Functions and Integrals and Circles

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1. Let P1 and P2 be two points on the unit sphere x2 + y2 + z2 = 1, and w and w2 the corresponding points on the plane
z = 0 under stereographic projection. Show that if P1 and P2 are antipodal points on the sphere, then W1W2 = ?1.
2. The hyperbolic functions sinh and cosh are defined
...sinhz = 2 coshz = 2...
Show that they satisfy the identities
sinh(iz) = isinz, cos(iz) = coshz.
3. Evaluate the integral where m and n are integers and C is the unit circle Izi = 1 taken counterclockwise.
4. Find the value of the integral of f(z) about the simple closed contour I z ? = 2 in the positive sense when
...
5. Expand 1/z2 in powers of z ? 1. Hint: You may find it easier to first solve the similar problem for 1/z and then differentiate.
6. Show that if 0 < b < a
....
7. (Extra for experts) Let C be the contour described by going from the origin along the x-axis to R, then around the circle of radius 1-? centred at the origin to Re2''3 (i.e. one third of the way around), and then straight back to the origin. Use this contour to show
....