Yes, nothing comes to mind quickly for you to use the Cauchy Integral formula. It should be fine using residues though.jiho.j said:I tried to set g(z) that is analytic inside C but I cannt set it.
Do I have to use Laurent seires or residue or something?
Integrating along C means that we are calculating the integral of the given function along a specific contour or path in the complex plane. In this case, C refers to the contour of a semicircle centered at the origin and passing through the points π/2 and -π/2.
The integral of tan(z/2) can be solved using the substitution method, where u = z/2 and du = dz/2. This simplifies the integral to ∫ tan(u)/2 du, which can then be solved using the trigonometric identity tan(u) = sin(u)/cos(u).
The singularities at z = π/2 and z = -π/2 can be handled by using the Cauchy Residue Theorem. This theorem states that the integral of a function along a closed contour is equal to the sum of the residues of the singularities within the contour. The residues at these singularities can be calculated and added to the integral to get the final result.
No, complex analysis techniques are necessary to solve this integral since it involves a contour integral and complex functions. Traditional integration techniques used in real analysis do not apply to complex functions.
The final result of solving this integral is a complex number. The exact value of this number will depend on the contour chosen and the singularities of the function. The result can be simplified further using trigonometric identities and other techniques.