(a) Suppose that a function f (z) is continuous on a smooth arc C, which has a parametric representation z = z(t) (a ‰¤ t ‰¤ b); that is, f [z(t)] is continuous on the interval a ‰¤ t ‰¤ b. Show that if φ(Ï„) (α ‰¤ Ï„ ‰¤ β) is the function described in Sec. 39, then

where Z(τ) = z[φ(τ)].
(b) Point out how it follows that the identity obtained in part (a) remains valid when C is any contour, not necessarily a smooth one, and f (z) is piecewise continuous on C. Thus show that the value of the integral of f (z) along C is the same when the representation z = Z(Ï„) (α ‰¤ Ï„ ‰¤ β) is used, instead of the original one.

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