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Jul9-11, 09:32 PM
P: 621
Quote Quote by TylerH View Post
[tex]\int_0^x darctanx=\int_0^x \frac{dx}{x^2+1}=\int_0^x \frac{i}{2(x-i)} - \frac{i}{2(x+i)} dx = \frac{i}{2} ln \left( \frac{x-i}{x+i} \right) \Rightarrow \forall x \in \Re, \: arctanx = \frac{i}{2}ln \left( \frac{x-i}{x+i} \right)[/tex]

Suppose we consider the integral[tex]\int\frac{1}{x-i}{dx}[/tex]

The path of integration has not been specified
[Interestingly the integrand does not satisfy the CR equations[Cauchy-Riemann equations] and hence it is not an analytical function. The derivative cannot be defined uniquely at any particular point.But given a path/route we should be able to define the derivative uniquely for points on the given path[by taking the tangential direction] rendering the integral suitable for the process of integration.Could anybody confirm or de-confirm the last statement?]

Sorry for the mistake.One may replace x by z=x+iy and specify the path along the x-axis.The CR equations do hold.