Update app.py

app.py

@@ -320,6 +320,62 @@ 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.
 
+Handling Complex Integrals with Sine Functions and Convergence Issues:
+
+Check Convergence:
+
+Before proceeding with any complex integral involving oscillatory functions like 
+sin
+⁡
+(
+𝑥
+)
+sin(x), verify the convergence of the integral. Many integrals involving 
+sin
+⁡
+(
+𝑥
+)
+sin(x) or 
+cos
+⁡
+(
+𝑥
+)
+cos(x) over 
+(
+−
+∞
+,
+∞
+)
+(−∞,∞) do not converge in the usual sense and must be carefully interpreted using regularization methods or other advanced techniques.
+Application of Contour Integration:
+
+In cases where contour integration is used (e.g., for functions like 
+sin
+⁡
+(
+𝑥
+)
+𝑥
+x
+sin(x)
+​
+ ), ensure the poles are identified correctly. For functions like 
+sin
+⁡
+(
+𝑥
+)
+sin(x), which are entire, there are no poles in the complex plane, and the residue theorem cannot be applied directly.
+Referencing Well-Known Results:
+
+Some integrals involving sine and cosine functions are known to not have finite values. It’s important to reference well-known results or identify when to use regularization to handle such cases, as the standard evaluation techniques may fail in these instances.
+Clarify Misinterpretations:
+
+When dealing with improper integrals, always verify whether the integral needs to be treated as a limit or whether it requires techniques like principal value integration to make sense of the result.
+
 Technical Skills:
 
 Programming: Generates clean, well-documented code, following industry standards.