082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

598 Chapter 19 Ordinary differential equations I
Jacobi-Anger identities
∞∑
e jxcos θ = J n
(x)j n e jnθ
n=−∞
∞∑
e jxsinθ = J n
(x)e jnθ
n=−∞
To obtain the first identity we let t = je jθ , so that t n = j n e jnθ , in Equation 19.36.
Noting that
t − 1 t = jejθ − 1
je jθ = jejθ +je −jθ = 2jcos θ
(where we have used the Euler relation cosθ = ejθ +e −jθ
, Section 9.7, p 339) then the
2
left hand sideof Equation 19.36 is
e x 2 (t−1 t ) = e x 2 (2jcosθ) = e jxcosθ
and sofrom Equation 19.36 we have
∞∑
e jxcosθ = J n
(x)j n e jnθ .
n=−∞
Inasimilarfashion, welett = e jθ , sothatt n = e jnθ , inEquation 19.36. Noting that
t − 1 t =ejθ − 1
e jθ = ejθ −e −jθ = 2jsin θ
(where we have used the Euler relation sinθ = ejθ −e −jθ
, Section 9.7 p 339) then the
2j
left hand sideof Equation 19.36 is
e x 2 (t−1 t ) = e x 2 (2jsinθ) = e jxsin θ
and sofrom Equation 19.36 we have
∞∑
e jxsinθ = J n
(x)e jnθ .
n=−∞
Engineeringapplication19.13
UseofBesselfunctionsinfrequencymodulation
A common type of modulation scheme used in electrical engineering is frequency
modulation(FM).Thepurposeofmodulationistoprepareasignalfortransmission,
such as when a signal is to be sent from a transmitter to a remote receiver during a
radiobroadcast.Atthereceiverthesignalisrecoveredfromitsmodulatedformbya
device called a demodulator.
Even though the process of electronically producing an FM signal is fairly
straightforward, the mathematics underpinning frequency modulation can become
complicatedandrequirestheuseofBesselfunctionsandaJacobi-Angeridentity.In