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Re: expansion of a complex exponential in terms of Bessel functions

For the case

ruleA=Exp[I x_a Cos[y_]]->BesselJ[0,xa]+I^n BesselJ[n,xa](Exp[Iny]+Exp[-Iny]))
Evaluate[Exp[I 3.14 a Cos[2 z]//.ruleA

I do not get an expansion in terms of Bessel functions; instead I get

Exp[3.14 I a Cos[2z]]

However, if I leave out the Complex I from the exponential,

ruleA=Exp[x_a Cos[y_]]->BesselJ[0,xa]+I^n BesselJ[n,xa](Exp[Iny]+Exp[-Iny]))
Evaluate[Exp[3.14 a Cos[2 z]//.ruleA

then I get

BesselJ[0,3.14a]+I^n (Exp[-2 I n z]+Exp[2 I n z])BesselJ[n, 3.14 a]

How can I get around this problem?



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