Yes.thomas49th said:Homework Statement
I don't understand why it's so great m>0 here
http://gyazo.com/cfa21b39653275a38ec80722b7faa58b
From these lectures notes (I think they are generally pretty good)
Where has the sin(mz) gone?
When going from 4.67 to 4.68 has
e^imz = cos(mz) + isin(mz) been used?
thomas49th said:I'm guessing that was the trick, but I don't see where sin(mz) has pootled off to?
thomas49th said:Homework Statement
I don't understand why it's so great m>0 here
http://gyazo.com/cfa21b39653275a38ec80722b7faa58b
From these lectures notes (I think they are generally pretty good)
Where has the sin(mz) gone?
When going from 4.67 to 4.68 has
e^imz = cos(mz) + isin(mz) been used? I'm guessing that was the trick, but I don't see where sin(mz) has pootled off to?
Jordan's Lemma is a theorem in complex analysis that describes the behavior of integrals with complex valued functions. It states that if a function approaches zero as a variable approaches infinity, then the integral of the function will also approach zero. In the case of sin(mz), the function approaches zero as z approaches infinity, thus the integral of sin(mz) will also approach zero.
One of the main technicalities associated with Jordan's Lemma and sin(mz) is the need for the function to be continuous and differentiable in the region of integration. If the function is not continuous or differentiable, then Jordan's Lemma cannot be applied. Additionally, the function must also approach zero as the variable approaches infinity in order for the theorem to be valid.
Yes, there are limitations to Jordan's Lemma when applied to sin(mz). One limitation is that the theorem only applies to integrals with real valued functions. This means that the imaginary component of sin(mz) must be zero in order for Jordan's Lemma to be applicable. Another limitation is that the function must also be continuous and differentiable in the region of integration, as mentioned previously.
Yes, Jordan's Lemma can be applied to other trigonometric functions, such as cos(mz) or tan(mz). As long as the function approaches zero as the variable approaches infinity and is continuous and differentiable in the region of integration, the theorem can be applied. However, the function must also have a real value, as Jordan's Lemma only applies to integrals with real valued functions.
Jordan's Lemma and sin(mz) have important implications in complex analysis. They allow for the evaluation of complex integrals by transforming them into real integrals, making them easier to solve. They also provide a way to evaluate integrals that would otherwise be difficult or impossible to solve using other methods. Overall, Jordan's Lemma and sin(mz) are powerful tools in the field of complex analysis.