Re: Complex bessel function

• To: mathgroup at smc.vnet.net
• Subject: [mg75186] Re: Complex bessel function
• From: dimitris <dimmechan at yahoo.com>
• Date: Fri, 20 Apr 2007 00:31:50 -0400 (EDT)
• References: <f079np\$46c\$1@smc.vnet.net>

\$VersionNumber
5=2E2

In[1]:=
f[a_, b_] := BesselI[0, a*Sqrt[I] + b]

In[2]:=
ComplexExpand[f[1, 2]]

Out[2]=
I*Im[BesselI[0, 2 + (-1)^(1/4)]] + Re[BesselI[0, 2 + (-1)^(1/4)]]

In[4]:=
(ComplexExpand[#1, TargetFunctions -> {Abs, Arg}] & )[f[1, 2]]

Out[4]=
Abs[BesselI[0, 2 + (-1)^(1/4)]]*Cos[Arg[BesselI[0, 2 + (-1)^(1/4)]]] +
I*Abs[BesselI[0, 2 + (-1)^(1/4)]]*
Sin[Arg[BesselI[0, 2 + (-1)^(1/4)]]]

In[3]:=
ComplexExpand[f[a, b]]

Out[3]=
I*Im[BesselI[0, (-1)^(1/4)*a + b]] + Re[BesselI[0, (-1)^(1/4)*a + b]]

In[5]:=
(ComplexExpand[#1, TargetFunctions -> {Abs, Arg}] & )[f[a, b]]

Out[5]=
Abs[BesselI[0, (-1)^(1/4)*a + b]]*Cos[Arg[BesselI[0, (-1)^(1/4)*a +
b]]] + I*Abs[BesselI[0, (-1)^(1/4)*a + b]]*
Sin[Arg[BesselI[0, (-1)^(1/4)*a + b]]]

???

=CF/=C7 Grasley, Zachary =DD=E3=F1=E1=F8=E5:
> I am currently trying to isolate the real and imaginary components of
> first and second order modified Bessel functions.  For example, I have a
> function expressed as
>
>
>
> F=BesselI[0,sqrt(i)*a+b], where i is the imaginary unit.  Both a and b
> are positive real.  I need to separate out the real and imaginary
> components of F symbolically.  I have tried Re, Im, ComplexExpand, I
> have used Assuming to define a and b as real and 0<a<infinity and
> 0<b<infinity.  I have tried ComplexExpand.  I have tried Refine[Im[F]].
> I loaded the package ReIm.
>
>
>
> All of these functions do a fine job separating the real and imaginary
> components when I provide numeric values for a and b, but are unable to
> solve symbolically.  I thought Refine was supposed to solve symbolically
> as if a numeric value has been assigned to the variables (with the value
> range of the variable defined by Assuming or the assumptions listed in
> Refine)?  Am I missing some simple trick to solve this symbolically, or
> is this not possible?
>
>
>