Is it possible to use complex logarithm to integrate 1/(z+i) along a path? Evaluate the following on the path gamma_1 with endpoints [−1,1+i] I_1=(i)/(2)int_(gamma_1) (1)/(z+i)dz -(i)/(2)int_(gamma_1)(1)/(z-i}dz Am I allowed to use the antiderivative Log(z+i) and Log(z−i) respectively to evaluate I_1, or the am I restricted by the complex logarithm's analyticity problems.
Jamarcus Schroeder
Open question
2022-08-19
Is it possible to use complex logarithm to integrate along a path? Evaluate the following on the path with endpoints
Am I allowed to use the antiderivative and respectively to evaluate , or the am I restricted by the complex logarithm's analyticity problems.
Answer & Explanation
elgrupomentasb
Beginner2022-08-20Added 9 answers
Yes, you are allowed to do what you want if you can find a branch of the logarithm that is holomorphic on a neighborhood of your curve. For example, if is a smooth curve from to along which is holomorphic, then , and
Of course you cannot do this if the logarithm takes a jump on the path, because then the function is no longer continuously differentiable. But you're allowed to use a different brach of for one of your integrals and another branch of the other. The justification is found in the Fundamental Theorem of Calculus for integrals of one parameter when you have an anti-derivative which is smooth on the curve.