MHB Finding Bounds for a Tricky Complex Integral

[Dustinsfl]

$$
\int_{\gamma}ze^{z^2}dz
$$

$\gamma(t) = 2t + i -2ti$, for $0\leq t\leq 1$.

$
\int_{\gamma} f(\gamma(t))\gamma'(t)dt
$

But

$
\int_{\gamma}ze^{z^2}dz \Rightarrow \frac{1}{2}\int e^wdw
$

So then I would be solving

$$
\frac{1}{2}\int\exp(4t-1+4ti-8t^2i)(4+4i-16ti)dw
$$

Correct? And how would I find the appropriate bounds for this integral or would it still be 0 and 1?

 

[Opalg]

$$
\int_{\gamma}ze^{z^2}dz
$$

$\gamma(t) = 2t + i -2ti$, for $0\leq t\leq 1$.

$
\int_{\gamma} f(\gamma(t))\gamma'(t)dt
$

But

$
\int_{\gamma}ze^{z^2}dz \Rightarrow \frac{1}{2}\int e^wdw
$

So then I would be solving

$$
\frac{1}{2}\int\exp(4t-1+4ti-8t^i)(4+4i-16ti)dw
$$

Correct? And how would I find the appropriate bounds for this integral or would it still be 0 and 1?

As you correctly say, $\int ze^{z^2}dz = \frac{1}{2}\int e^wdw = \frac12e^w = \frac12e^{z^2}$. In other words, your function has an indefinite integral. The fundamental theorem of integration applies, and says that the value of the integral along the path $\gamma$ is the difference between the values of the indefinite integral at the endpoints of $\gamma$.

The endpoints of $\gamma$ are $\gamma(0) = i$ and $\gamma(1) = 2-i$. So the integral is equal to $\frac12\left(e^{(2-i)^2} - e^{i^2}\right)$.

 

[Dustinsfl]

As you correctly say, $\int ze^{z^2}dz = \frac{1}{2}\int e^wdw = \frac12e^w = \frac12e^{z^2}$. In other words, your function has an indefinite integral. The fundamental theorem of integration applies, and says that the value of the integral along the path $\gamma$ is the difference between the values of the indefinite integral at the endpoints of $\gamma$.

The endpoints of $\gamma$ are $\gamma(0) = i$ and $\gamma(1) = 2-i$. So the integral is equal to $\frac12\left(e^{(2-i)^2} - e^{i^2}\right)$.

So you are evaluating this integral
$$
\frac{1}{2}\int_{i}^{2-i}\exp\left(4t-14ti-8t^2i\right)(2-2i)dw
$$

 

[polygamma]

So you are evaluating this integral
$$
\frac{1}{2}\int_{i}^{2-i}\exp\left(4t-14ti-8t^2i\right)(2-2i)dw
$$

No, he's evaluating $\displaystyle \int_{i}^{2-i} z e^{z^{2}} \ dz$. The path taken doesn't matter.

I'll state the theorem directly from my old textbook.

Let $f$ be analytic in a simply connected domain D. If $z_{0}$ and $z_{1}$ are any two points in D joined by a contour $C$, then $\displaystyle \int_{C} f(z) \ dz = \int_{z_{0}}^{z_{1}}f(z) \ dz= F(z_{1})-F(z_{0})$ where $F$ is an antiderivative of $f$ in $D$.

$f(z) = ze^{z^{2}}$ is an entire function. It's analytic everywhere. So the above applies.

 

[Dustinsfl]

No, he's evaluating $\displaystyle \int_{i}^{2-i} z e^{z^{2}} \ dz$. The path taken doesn't matter.

I'll state the theorem directly from my old textbook.

Let $f$ be analytic in a simply connected domain D. If $z_{0}$ and $z_{1}$ are any two points in D joined by a contour $C$, then $\displaystyle \int_{C} f(z) \ dz = \int_{z_{0}}^{z_{1}}f(z) \ dz= F(z_{1})-F(z_{0})$ where $F$ is an antiderivative of $f$ in $D$.

$f(z) = ze^{z^{2}}$ is an entire function. It's analytic everywhere. So the above applies.

After I make the substitution, I have $\displaystyle\frac{1}{2}\int_i^{2-i}e^wdw$ what is $w = $ now?

 

[polygamma]

$\displaystyle F(z) = \int f(z) = \int ze^{z^{2}} \ dz = \frac{1}{2} e^{z^{2}} + C $

so $\displaystyle \int_{\gamma} f(z) \ dz = \int_{i}^{2-i} f(z) \ dz = F(2-i) - F(i) = \frac{1}{2}e^{(2-i)^{2}} - \frac{1}{2} e^{i^{2}}$ and then simplify if you want

 

[Dustinsfl]

$\displaystyle F(z) = \int f(z) = \int ze^{z^{2}} = \frac{1}{2} e^{z^{2}} + C $

so $\displaystyle \int_{\gamma} f(z) \ dz = \int_{i}^{2-i} f(z) \ dz = F(2-i) - F(i) = \frac{1}{2}e^{(2-i)^{2}} - \frac{1}{2} e^{i^{2}}$ and then simply if you want

That is what I wasn't understanding.