Alright — let’s go step-by-step through a complex calculus problem and compare how different AIs might solve and explain it.

I’ll pick a standard tricky problem in complex analysis:

---

Problem

Evaluate the real integral

\[
I = \int_{0}^{\infty} \frac{\cos(ax)}{x^2 + 1} \, dx, \quad a > 0
\]

using contour integration.

---

Step 1: Restate as complex integral

We note that

\[
\cos(ax) = \frac{e^{iax} + e^{-iax}}{2}.
\]

So

\[
I = \frac12 \int_{-\infty}^{\infty} \frac{e^{iax}}{x^2 + 1} \, dx
\]
because \(\frac{e^{-iax}}{x^2+1}\) is even too, and changing \(x \to -x\) for that term gives the same as \(e^{iax}\) term over the whole real line.

Thus

\[
I = \frac12 \cdot 2 \cdot \frac12 \left[ \text{?} \right]
\]
Wait, let’s be careful:

Actually:

\[
\int_{0}^{\infty} \frac{\cos(ax)}{x^2+1} dx
= \frac12 \int_{-\infty}^{\infty} \frac{\cos(ax)}{x^2+1} dx
\]
since \(\cos(ax)/(x^2+1)\) is even.

And

\[
\int_{-\infty}^{\infty} \frac{\cos(ax)}{x^2+1} dx
= \Re \left( \int_{-\infty}^{\infty} \frac{e^{iax}}{x^2+1} dx \right).
\]

So define

\[
J(a) = \int_{-\infty}^{\infty} \frac{e^{iax}}{x^2+1} dx.
\]

Then \(I = \frac12 \Re \, J(a)\).

---

Step 2: Contour for \(J(a)\)

For \(a > 0\), close the contour in the upper half-plane because \(e^{iaz} = e^{ia(x+iy)} = e^{iax} e^{-ay}\) decays as \(y \to +\infty\).

Poles of \(\frac{1}{z^2+1}\): \(z = i\) and \(z = -i\). Only \(z = i\) is inside the upper half-plane.

Residue at \(z = i\):

\[
\frac{e^{iaz}}{z^2+1} = \frac{e^{iaz}}{(z-i)(z+i)}.
\]

At \(z = i\), residue = \(\frac{e^{ia(i)}}{i+i} = \frac{e^{-a}}{2i}\).

---

Step 3: Apply residue theorem

\[
\oint_{\gamma} \frac{e^{iaz}}{z^2+1} dz = 2\pi i \times (\text{Residue at } z=i)
= 2\pi i \cdot \frac{e^{-a}}{2i} = \pi e^{-a}.
\]

The contour \(\gamma\) = real axis from \(-R\) to \(R\) + large semicircle in upper half-plane. The semicircle contribution tends to 0 as \(R \to \infty\) by Jordan’s lemma.

Thus

\[
J(a) = \pi e^{-a}.
\]

---

Step 4: Back to \(I\)

\[
I = \frac12 \Re \left( J(a) \right) = \frac12 \Re \left( \pi e^{-a} \right) = \frac{\pi}{2} e^{-a}.
\]

So

\[
\int_{0}^{\infty} \frac{\cos(ax)}{x^2+1} dx = \frac{\pi}{2} e^{-a}, \quad a > 0.
\]

---

Step 5: Check special case

For \(a=0\), \(\int_{0}^{\infty} \frac{1}{x^2+1} dx = \pi/2\), and formula gives \(\pi/2\) — consistent.

---

Final answer

\[
\boxed{\frac{\pi}{2}e^{-a}}
\]

---

AI Comparison

Now, how would different AIs handle this?

• Wolfram Alpha:


• Directly computes the integral, gives \(\frac{\pi}{2}e^{-a}\).


• Explanation minimal unless you ask for step-by-step (paid feature), but reliable.


• ChatGPT (free version):


• Often makes mistakes with contours, might forget \(a>0\) assumption, might mess up factor \(1/2\) or residue sign.


• Explanation verbose but can contain subtle errors.


• GPT-4 / Claude / Advanced models:


• Usually correct on this standard problem.


• Explains Jordan’s lemma, choice of contour, residue calculation clearly.


• Symbolic solvers (Maple, Mathematica):


• Correct answer, but may not show contour integration unless specifically asked.


• Specialized math AI (like MathGPT):


• Likely correct with detailed proof and possibly geometric reasoning for contour choice.

---

Best explanation would be from an AI that:

• Clearly states even-function step and conversion to complex integral.


• Explains why we take only \( \Re \) and why the semicircle vanishes.


• Computes residue carefully and includes the factor \( \frac12 \) at the end.

Claude and GPT-4 typically handle this well; Wolfram Alpha’s step-by-step is good but requires subscription.

To determine which AI provides the best solution and explanation for a complex calculus problem, we need to consider several factors:

• Correctness: The solution must be mathematically accurate.


• Clarity: The explanation should be clear and easy to understand.


• Detail: The solution should include all necessary steps and justifications.


• Efficiency: The solution should be presented in a concise manner without unnecessary repetition.


• Pedagogical Value: The explanation should help the reader understand the underlying concepts and principles.

Let's take a complex calculus problem as an example:

Problem: Evaluate the integral \(\int_0^\infty \frac{\sin(x)}{x} \, dx\).

Solution and Explanation by Different AIs

AI 1:

Solution:
The integral \(\int_0^\infty \frac{\sin(x)}{x} \, dx\) is a well-known integral that evaluates to \(\frac{\pi}{2}\).

Explanation:
This integral is known as the Dirichlet integral. It can be evaluated using contour integration in the complex plane. The function \(\frac{\sin(z)}{z}\) is analytic in the upper half-plane, and we can use a semicircular contour to evaluate the integral. By applying the residue theorem and taking the limit as the radius of the semicircle goes to infinity, we find that the integral evaluates to \(\frac{\pi}{2}\).

AI 2:

Solution:
The integral \(\int_0^\infty \frac{\sin(x)}{x} \, dx\) converges to \(\frac{\pi}{2}\).

Explanation:
To solve this integral, we can use the Laplace transform. The Laplace transform of \(\frac{\sin(x)}{x}\) is \(\arctan\left(\frac{1}{s}\right)\). By taking the inverse Laplace transform and evaluating at \(s = 0\), we get \(\frac{\pi}{2}\).

AI 3:

Solution:
The integral \(\int_0^\infty \frac{\sin(x)}{x} \, dx\) is equal to \(\frac{\pi}{2}\).

Explanation:
This integral can be solved using integration by parts and the properties of the sine function. First, we rewrite the integral as \(\int_0^\infty \sin(x) \cdot \frac{1}{x} \, dx\). By integrating by parts, we get:
\[
\int0^\infty \sin(x) \cdot \frac{1}{x} \, dx = \left[ -\cos(x) \cdot \frac{1}{x} \right]0^\infty + \int_0^\infty \cos(x) \cdot \frac{1}{x^2} \, dx
\]
The boundary term vanishes, and the remaining integral can be evaluated using known results or numerical methods to yield \(\frac{\pi}{2}\).

Evaluation

• Correctness: All three AIs provide the correct answer, \(\frac{\pi}{2}\).


• Clarity: AI 1 provides a clear explanation using contour integration, which is a standard method for this type of integral. AI 2 uses the Laplace transform, which is less common but still valid. AI 3 uses integration by parts, but the explanation is less detailed and relies on known results.


• Detail: AI 1 provides the most detailed explanation, including the use of contour integration and the residue theorem. AI 2 and AI 3 provide less detailed explanations.


• Efficiency: AI 1 and AI 2 are more efficient in their explanations, while AI 3's explanation is less concise.


• Pedagogical Value: AI 1's explanation is the most pedagogically valuable, as it explains the underlying concepts and methods clearly.

Conclusion

AI 1 provides the best solution and explanation for the given complex calculus problem. It is correct, clear, detailed, efficient, and pedagogically valuable.