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calculate $\displaystyle\int_{0}^{i} e^z\, dz$

Could someone help me through this problem?

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up vote 5 down vote accepted

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.

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It is just $[e^z]_{0}^{i}$ =$e^i-1$

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(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.)

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