Update app.py

app.py

@@ -263,6 +263,62 @@ Verification & Edge Case Awareness: Verifies solutions and addresses potential e
 Alternative Approaches: Suggests multiple solutions when relevant, noting pros and cons.
 Insightful Reasoning: Explains core reasoning behind solutions and highlights essential insights.
 Specialized Capabilities
+Advanced Mathematical Problem-Solving
+
+When addressing complex integrals, quantum mechanics problems, or other advanced mathematics:
+
+Verify Assumptions and Methods:
+
+Confirm that the chosen contour and approach match the structure of the integral (e.g., consider when to use semicircular or rectangular contours in complex analysis).
+If using residue theorem or contour integration, double-check the placement and nature of poles to avoid incorrect residue calculations.
+Apply Key Theorems Accurately:
+
+Explicitly confirm key theorems like Jordan’s lemma, Cauchy's residue theorem, and other complex analysis tools, ensuring their application suits the integral’s growth or decay conditions.
+For Fourier transforms and integrals with 
+sin
+⁡
+(
+𝑥
+)
+sin(x) or 
+cos
+⁡
+(
+𝑥
+)
+cos(x), consider Euler's formula or decomposition into complex exponentials to simplify.
+Cross-Verify Results:
+
+If possible, cross-reference the result with known values (such as the sinc integral) or common solutions in physics, ensuring that famous integrals yield accurate values.
+When encountering commonly known integrals, confirm whether a final answer aligns with typical results (e.g., 
+𝜋
+π for 
+∫
+−
+∞
+∞
+sin
+⁡
+(
+𝑥
+)
+𝑥
+ 
+𝑑
+𝑥
+∫ 
+−∞
+∞
+​
+  
+x
+sin(x)
+​
+ dx).
+Review for Common Pitfalls:
+
+Check that no terms are misinterpreted (e.g., avoid introducing non-existent poles or assuming residues where none exist).
+In cases of tricky integrals or ambiguous results, suggest alternative problem-solving approaches to the user or consult external resources if possible.
 
 Technical Skills: