Finding the Primitive of a Complex Integral

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SUMMARY

The discussion focuses on finding the primitive of the complex integral $\int_{\gamma}ze^{z^2}dz$ from $i$ to $2-i$. The solution involves recognizing that the derivative of $\frac{e^{z^2}}{2}$ is equal to $ze^{z^2}$, allowing the use of the Fundamental Theorem of Calculus. This theorem applies due to the analyticity of the function involved, confirming that the integral can be evaluated directly from the antiderivative.

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How can I find the primitive of $\int_{\gamma}ze^{z^2}dz$ from $i$ to $2-i$?
 
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dwsmith said:
How can I find the primitive of $\int_{\gamma}ze^{z^2}dz$ from $i$ to $2-i$?

$$\int z e^{z^{2}}\,dz=\frac{1}{2}\int 2z e^{z^{2}}\,dz.$$

Can you finish?
 
Ackbach said:
$$\int z e^{z^{2}}\,dz=\frac{1}{2}\int 2z e^{z^{2}}\,dz.$$

Can you finish?

So $\left(\frac{e^{z^2}}{2}\right)'=\int ze^{z^2}dz$ Then to solve the integral I just integrate g'(z) right?
 
dwsmith said:
So $\left(\frac{e^{z^2}}{2}\right)'=\int ze^{z^2}dz$ Then to solve the integral I just integrate g'(z) right?

Actually, I would have said that

$$\left(\frac{e^{z^{2}}}{2}\right)'=ze^{z^{2}}.$$

Then just use the Fundamenal Theorem of the Calculus, which works because your function is analytic.
 

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