Finding the Primitive of a Complex Integral

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Discussion Overview

The discussion revolves around finding the primitive (antiderivative) of the complex integral $\int_{\gamma}ze^{z^2}dz$ from $i$ to $2-i$. Participants explore different approaches to solving this integral, including the application of the Fundamental Theorem of Calculus and properties of analytic functions.

Discussion Character

  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant asks how to find the primitive of the integral $\int_{\gamma}ze^{z^2}dz$.
  • Another participant suggests rewriting the integral as $\frac{1}{2}\int 2z e^{z^{2}}\,dz$ and asks for assistance in completing the solution.
  • A participant states that $\left(\frac{e^{z^2}}{2}\right)'=\int ze^{z^2}dz$ and questions whether integrating $g'(z)$ is the correct approach.
  • Another participant confirms that $\left(\frac{e^{z^{2}}}{2}\right)'=ze^{z^{2}}$ and mentions using the Fundamental Theorem of the Calculus, noting that the function is analytic.

Areas of Agreement / Disagreement

Participants express different methods for approaching the integral, and while there is some agreement on the use of the Fundamental Theorem of Calculus, the discussion does not reach a consensus on the best approach to solve the integral.

Contextual Notes

There are unresolved steps in the integration process, and the discussion does not clarify the specific path of integration or the implications of the chosen methods.

Dustinsfl
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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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