Finding the Primitive of a Complex Integral

[Dustinsfl]

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

 

[Ackbach]

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?

 

[Dustinsfl]

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

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?

 

[Ackbach]

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.

 

[blue_raver22]

The primitive of a complex integral can be found by using the Fundamental Theorem of Calculus for Complex Functions, which states that the primitive of a complex function is the antiderivative of the function. In this case, the antiderivative of $ze^{z^2}$ is simply $e^{z^2}$.

To find the primitive of the given integral from $i$ to $2-i$, we can use the following steps:

1. Rewrite the integral using the antiderivative: $\int_{\gamma}ze^{z^2}dz = \left[e^{z^2}\right]_{\gamma}$.

2. Substitute the upper limit of integration, $2-i$, into the antiderivative: $e^{(2-i)^2}$.

3. Substitute the lower limit of integration, $i$, into the antiderivative: $e^{i^2} = e^{-1}$.

4. Subtract the result from step 3 from the result of step 2: $e^{(2-i)^2} - e^{-1}$.

Therefore, the primitive of the given integral from $i$ to $2-i$ is $e^{(2-i)^2} - e^{-1}$.