complex analysis Notes

1)  L Hospital’s  rule is not valid for complex functions.

Consider f(x) = √x, g1(x) = √x + x sin(1/x), g2(x) = √x + x. cos(1/x), g(x) =g1(x) + ig2(x)

 Then limxà0 f(x) = 0 , lim xà0 g(x) = 0.

Lim xà0 f(x)/g(x) = 1/(1+i) ,but   Lim xà0 (df/dx) / (dg/dx) = 0.

2) am. bm = (ab)m . The rule is not valid in complex.

Let w = √i  = e iπ/4 then   wi.(i)i =  (√-i)i.  Lhs. = e-π/4 . e-π/2 = e-3π/4

Rhs = eπ/2.