calculate $\displaystyle\int_{0}^{i} e^z\, dz$
Could someone help me through this problem?
calculate $\displaystyle\int_{0}^{i} e^z\, dz$
Could someone help me through this problem?
(1) Parametrize the line segment from $0$ to $i$ in $\Bbb C$, ie $\gamma:[0,1]\to C$, where $C$ is the line segment.
(2) Write $\displaystyle \int_C e^z dz=\int_0^1 \big(\exp\gamma(t)\big)\gamma\,'(t)dt$ and compute it using familiar calculus.
(The fundamental theorem of calculus actually applies in the complex plane too, but I assume you're looking for a nitty-gritty approach given it's homework. Note for this approach, $e^z$ is entire.)
It is just $[e^z]_{0}^{i}$ =$e^i-1$
The function $e^z$ is entire. Hence, contour integrals are independent of path. The antiderivitive of $e^z$ is just $e^z$. Stick in the endpoints, and do the subtraction.