Question

Jordan's Lemma? If anyone here actually knows what it is, help would be appreciated.

Answer 1

I don't understand it :(

Answer 2

I just want to know what it is because I need it for the midterm

Answer 3

@Directrix please help if you can

Answer 4

Sorry to say: I have not studied it yet.

Answer 5

Its okay. Thanks for being honest

Answer 6

I have a good knowledge about the Jordan's Lemma @Thefaceless

Answer 7

can you please explain :)

Answer 8

please suppose that we have to compute the subsequent integral:
\[\int\limits_{ - \infty }^{ + \infty } {f(x){e^{i\nu x}}} dx,\quad \nu \in \mathbb{R}\]

Answer 9

and f(x) can be extended to the complex plane, in order to get a holomorphic function f(z)

Answer 10

then we can rewrite that integral as follows:
\[\int\limits_{ - \infty }^{ + \infty } {f(z){e^{i\nu z}}} dz\]

Answer 11

ok

Answer 12

ok!, now we can demonstrate, that we have the subsequent result:
\[\begin{gathered}
\int\limits_{ - \infty }^{ + \infty } {f(x){e^{i\nu x}}} dx = 2\pi i\sum\limits_{\operatorname{Im} {z_i} > 0} {{R_f}\left( {{z_i}} \right)} \quad if\;\nu > 0 \hfill \\
\int\limits_{ - \infty }^{ + \infty } {f(x){e^{i\nu x}}} dx = - 2\pi i\sum\limits_{\operatorname{Im} {z_i} < 0} {{R_f}\left( {{z_i}} \right)} \quad if\;\nu < 0 \hfill \\
\end{gathered} \]

Answer 13

okay i see that

Answer 14

where:
\[{R_f}\left( {{z_i}} \right)\]
is the residual value of the function f(z) at z_i, namely:
\[{R_f}\left( {{z_i}} \right) = \frac{1}{{2\pi i}}\oint\limits_\gamma {f\left( z \right)dz} \]

Answer 15

being gamma a little circumference in the z-plane which encloses z_i

Answer 16

oops...circumference "gamma"

Answer 17

okay

Answer 18

nevertheless, it is necessary to see how the Jordan's Lemma works, so we can try to compute this integral:
\[\int\limits_{ - \infty }^{ + \infty } {\frac{{{e^{ix}}}}{{x\left( {{x^2} + 1} \right)}}} \;dx\]

Answer 19

wait we haven't shown it yet right?

Answer 20

this is just preliminary

Answer 21

yes!

Answer 22

then I'm going to demonstrate the Jordan's Lemma

Answer 23

okay

Answer 24

Let's consider the first case, namely: \[\nu > 0\]

Answer 25

then we have to consider an half-circumference which lieas in the upper half-plane, namely: