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

Please see the attached file for the fully formatted problems.
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
....

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Solution Summary

Complex Variables, Stereographic Projections, Hyperbolic Functions and Integrals and Circles are investigated.

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