Evaluate $\int_0^ie^z\ dz$.
I have never encountered an integral with limits in Complex Analysis yet. I am only familiar with the symbol $\int_C$ where $C$ is some path. What do the limits mean in this problem?
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Sign up to join this communityEvaluate $\int_0^ie^z\ dz$.
I have never encountered an integral with limits in Complex Analysis yet. I am only familiar with the symbol $\int_C$ where $C$ is some path. What do the limits mean in this problem?
If the function you are integrating is entire (analytic everywhere), then the integral does not depend on path and you can integrate just as you do for real functions, $$\int_\alpha^\beta f(z)\,dz=F(\beta)-F(\alpha)\ ,$$ where $F'(z)=f(z)$. In your case, $f(z)=e^z$ so $F(z)=e^z$ and $$\int_0^i e^z\,dz=e^i-e^0=\cos1+i\sin1-1\ .$$ There is more that can be said about this kind of problem, I am sure there will be theorems about it in your course.