The residue of a function at a point is the value of the coefficient of the term with the highest negative power in the Laurent series expansion around that point. In this case, the Laurent series expansion of exp(1/(z+i)) around z=-i is 1/(z+i). Therefore, the residue at z=-i is 1.
To find the residue of a function at a point, you can use the formula Res(f,z0) = lim(z→z0) (z-z0)f(z). In this case, the point z0 is -i, so the residue is Res(exp(1/(z+i)), -i) = lim(z→-i) (z+i)exp(1/(z+i)) = (1-i)exp(1/(z+i)) = 1.
This is because the coefficient of the term with the highest negative power in the Laurent series expansion of exp(1/(z+i)) around z=-i is 1/(z+i), which is equal to 1 when evaluated at z=-i. Therefore, the residue at z=-i is 1.
No, the residue of a function at a point cannot be negative. In this case, the residue is equal to 1, which is a positive value.
Yes, the residue at a point can be affected by the presence of poles and zeros in the function. However, in this case, exp(1/(z+i)) has a simple pole at z=-i, so the residue is not affected by any other poles or zeros.