A Tutorial on Bessel Functions

8/12/2019 A Tutorial on Bessel Functions

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A Tutorial on Bessel Functions andNumerical Evaluation Bessel Integrals UIUIlk and L Sevgi

D o g u ~ University, Electronics and Communications Engineering DepartmentZeamet Sokak 21, Acibadern - Kadikoy, 34722 Istanbul, TurkeyE-mail: [email protected]

AbstractThis paper aims to provide a tutorial on Bessel functions, and especial ly on the numerical evaluation of Bessel integrals.Bessel integrals are asymptotically evaluated using high-frequency methods, such as the stationary-phase method, steepestdescent path evaluations, and uniform asymptotics. Concepts such as saddle points and their contributions, and integrationover the steepest paths are emphasized. A MATLAB simulation package, BESSEL_GUI, was prepared to compare eachmethod with the other , and also with the MATLAB built-in function besse l j (u , z ) . The tool also allows plotting themagnitude of the complex integrand of the Bessel integral in three dimensions. The user can thus visualize the locations ofthe saddle points and the steepest-decent paths.Keywords: Bessel functions ; saddle points; contour deformations; stationary phase method; steepest descent path; MATLAB;graphical user interface; simulation ; three-dimensional graphics; visualization; engineering education, numerical analysis

z

Figure 1b. A wedge-shaped waveguide in two dimensions.y

PEe

P E C ~ __-t- /

x -------- --(p,tp)

Figure la . A ring resonator with rectangular cross section andthe cylindrical coordinates. The resonator is bounded radially,vertically, and horizontally between a p s b, 0 s z s h, andos (i 2T , respectively.

Hankel, Bessel, and Neumann functions reduce to simple exponential and trigonometric functions. The Hankel functions H ~ l zand H ~ 2 (z) correspond to incident and reflected (progressing)waves, while the Bessel and Neumann functions correspond tostanding waves.

1. Introduction

Normally, when a Bessel function is said a Bessel function ofthe first kind, Ju(z) is automatically meant. Certain physicalproblems have solutions in the form of combinations of Besselfunctions that are called Bessel functions of the second or thirdkind. The Bessel function of the second kind is also called theNeumann function, Nu (z). Bessel functions of the third kind arealso known as Hankel functions, H (z) and H ~ 2 (z).These functions are interrelated: J =(H(l) +H(2 ) /2.u uNu = H ~ l - H ~ 2 )/2i . Table I shows that for large arguments z,

B essel functions are defined as the solutions y (z) of the Bessel2differential equation z2 d + Z dy +( z2 _ u2 )Y = 0 for an

dz dzarbitrary real or complex number u, where u is the order of theBessel function [1-4]. The Bessel equation arises in the solution ofthe Helmholtz and Laplace equations in cylindrical (rather thanCartesian or spherical) coordinates. Bessel functions are thereforeespecially important for many physical phenomena of wave propagation and static potentials, including heat conduction or electricityflow in a solid cylinder, the propagation of electromagnetic wavesin a cylindrical waveguide, diffusion problems on a lattice, themotions of fluids, the diffraction of light, and the deformations ofelastic bodies. For example, the solutions to the ring resonator withrectangular cross section in Figure la [5, 6] and the solutions to thewedge-shaped waveguide in Figure 1b can be obtained in closedform in terms ofBessel functions [7].

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Table 1. Relationships among the Hankel, Bessel, andNeumann functions, and the exponential and trigonometric functions.Hankel H ~ l ( z i z ~ i l Z - IncidentFunction 7iZ WaveHankel H ~ 2 z e- i z ~ - i[z-(u+1I21T /2] ReflectedFunction -e Wave7iZBessel J u z cos(z) ~ cos[z -(u+ l/ 2)7i/2]Function 7iZ Standing

Neumann ~ sin[z - (u+ l/2)7i/2 ] WavesFunction Nu z sin(z) 7iZTables of values for Bessel functions can be found in manyfundamental mathematics books [I]. Moreover, Bessel functionscan also be defined by contour integrals. Four different types ofintegral representations of Bessel integrals are given in Section 2[2]. High-frequency asymptotic evaluations of these integrals usingmethods such as the stationary-phase method (SPM), steepestdecent path (SDP) evaluation, and uniform asymptotics (UA) aregiven in Section 3. Some important mathematical expressions,such as saddle points (SP) and contour deformations, are also pre

sented in Section 3.A MATLAB-based virtual tool, BESSEL_GUI, is introducedin Section 4. BESSEL_GUI computes values of the Bessel functionasymptotically using different types of Bessel integrals, and thencompares them with the MATLAB built-in function be s s e l j (u ,z}, where o is the order and z is the argument. The tool also

allows plotting the magnitude of the complex integrand of the

Im(y) er-t----,..---Re(y)

-1tFigure 2c. The integration contour for the Type-Ill Bessel integral representation.Im(t)

Re(t)

Figure 2a. The integration contour for the Type-I Bessel integral representation.

1

Im(w)1 Re(w)

-n

Im(a)

n Re(a)

Figure 2d. The integration contour for the Type-IV Besselintegral representation.Bessel function in three dimensions. The user can thus visualizelocations of the saddle points and also the steepest-decent paths.BESSEL_GUI can be downloaded from the http://www3.dogus.edu.tr/lsevgi (or /culuisik) Web site.

2. Integral Representations ofBessel FunctionsFigure 2b. The integration contour for the Type-II Bessel integral representation.IEEEAntennasandPropagation Magazine, Vol. 51, No. 6, December 2009

Bessel functions of any complex order u can be representedas integrals [2] as shown in Equation (I) . The integration contour,Ct , in Equation (I) is shown inFigure 2a.

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Transforming the t domain into an a domain, def ined asa = i In(t), the Bessel function can be represented as in Equa-tion (2) with the integration contour Ca ' shown in Figure 2b:

( 1O.5z t- -Jv(z)=-. fe t t-V-Idt.27rl C (1)3.1.1 Type I

For the Type I first-kind Bessel integral representation givenin Equation (1), the magnitude and phase functions become

(7a)

r;(z)= _1_ f e-izsin(a)+ivada .27rC

a(2) t - v- lf t)=- .27rl (7b)

Finally, transforming the a domain into a w domain, defined asw = sin(a ), the Bessel function can be represented as in Equa-tion (4) with the integration contour Cw shown in Figure 2d:

Similarly, transforming the a domain into a r domain, defined asr = - ia , the Bessel function can be represented as in Equation (3),with the integration contour Cy shown in Figure 2c:

Ju ( z)= _1_. fezsinh(y)-vydr .27rl Cr (3)The s ta tionary point s for the Type-I Bessel funct ion Ju ( z) arefound to be tsI = j and ts2 =- j . As shown in Figure 3a, these stationarypoints are located symmetrically along the imaginary taxis.They are isolated and independent of z and v. These stationarypoints therefore never coincide for different z and v values.

The contributions of the first s ta tionary point can be computed, using Equation (6), as

(8a)

(4) () z - j1f/2q ts l =-3=ze ,} (8b)

The contribution of the second stationary point can be computedusing Equation (6) as

3. Asymptotic Evaluations ofBessel Integrals3.1 Stationary-Phase Method SPM

j [ z + ~ . - : . V+I ]e 4 2JV\(z) ~ .27rz (8c)

The Bessel function can be represented as an integral with thecomplex magnitude function f (t) and the complex phase functionq(t) [3]:

The s ta tionary point s (SPs) are determined by the zeros of thederivative of the phase funct ion, q (t), with respect to t. Thestationary-phase method takes into account only the contributionsof the stationary points, and approximates the Bessel funct ion interms of the stationary-point contributions as

The Ju ( z) can then be approximated as the superposi tion of thetwo stationary-point contributions, JvI (z) and Jv2 ( z), as

(9a)

(9b)

(9c)

() z j 1l12q ts2 = - ( _ j)3 = ze ,(5)v(z)= feq(t)f(t)dt.

C

(6) Jv z =Jv\ z +Jv2 z = 2 cos[z+ :: _ :: V+1 )J.V;; 4 2(10)

The stationary points for four different types of integral representations for the first-kind Bessel functions, presented in Section 2,can be derived, and their contr ibut ions can be computed, as follows.

The Type-I Bessel function of order 0 J0 ( z , computed via thestationary-phase method, is plotted in Figure 4, and compared withthe MATLAB built-in be s s e l j function (Note that a two-line

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Im(t) Im(w)ts.... et

Re(t) -1 1 Re(w)CwFigure 3a. The integration contours, stationary points, andsteepest-decent paths for the Type-I Besselintegral representation. Figure 3d. The integration contours, stationary points, andsteepest-decent paths for the Type-IV Bessel integral representation.

Im(a)0.5

n Re(a )o

-0.4oFigure 3b. The integration contours, stationary points, andsteepest-decent paths for the Type-Il Bessel integral representation.

Im y

zFigure 4. The Type-I, first-kind Bessel function of order 0computed via the stationary-phase method and the MATLABbuilt-in besse l j function.

--MATLAB code plots the curve in the figure: z=O: 0 .0 5 : 50;p l o t ( z , besse l j (O , z ) ) ) . For v=O , the stationary-pointcontributions were adequate to approximate the Bessel function foreveryz except at z =v .

-7t

Rey

The Type-I first-kind Bessel functions of orders 3 and 6J3 ( z) and J6 ( Z) ) , computed via the stationary-phase method,are plotted in Figures Sa and Sb, respectively. Note that the stationary-phase method approximates the first-kind Bessel function onlyfor z v .

3.1.2 Type For the second Bessel integral representation given in Equa

tion (2), the magnitude and phase functions become

Figure 3c. The integration contours, stationary points, andsteepest-decent paths for the Type-Ill Besselintegral representation.IEEEAntennasandPropagation Magazine ,Vol. 51, No. 6,December 2009

q(t) = - izsin(t)+ivt,

1f t =- .2

(lla)

(lIb)

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The contribution of the second stationary point can be computed as

-0 . 4 :. jo 1 0 2 0 3 0 4 0

( ) - I - izsin(ls.)+ivIs1Jvt z ' e .27Z izsin(lsd (Bc)

(l4a)

(14b)Figure Sa. The Type-I, first-kind Bessel functions J3 (z) foroS; z S; 50 computed via the stationary-phase method and theMATLAB built-in besse l j function. (14c)

For z > v , the Bessel function can be approximated as the superposition of the two stationary-point contributions JvI (z) andJv2 ( z ) . Howeve r, for z


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Im(t)

Z=2, v 50

Re( t) ,'' 1

.41 0

Re(t)-234

50o

-50-100-150-200

logl[(t) eq(t)/200150100

Figure 6a. The magnitude of the complex integrand in theType-II Bessel-integral representation for the case ofz >v z=30, v=2 .

Figure 6b. The magnitude of the complex integrand in theType-II Bessel-integral representat ion for the case of z < v( z=2 , v =50).

0.2

305 = 10

f \: \ .....V

Uniform AsympteticsMatLab

.f \i \ . ....

;V15 20z

....

f \ II .

10

Type 4

~

-0.2

30\ ' = 4

255 20z

10

1 J v z

0.5

-0.4o

Figure 7. The Type-II Bessel function of order 4 computed viathe stationary-phase method, steepest-decent-path evaluation,and the MATLAB built-in besse l j function.

Figure 8. The Type-IV, first-kind Bessel function of order 10computed via uniform asymptotics and the MATLAB built-inbesse l j function.

= 425 30

I + Direct Integration I- MatLab

20z50

Type 2

. 5 r ; - ; ~ = = =Jv(z)

-0.3oFigure 9. The Type-II, first-kind Bessel function of order 4computed via direct numerical integration and the MATLABbuilt-in besse l j function.

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3.1.3 Type IIIFor the third Bessel integral representation given in Equation (3), the magnitude and phase functions become

The Type-IT, first-kind Bessel function of order 4 J4 (z),computed via the stationary-phase method, is plotted in Figure 7and compared with the MATLAB built-in besse l j function. Thestationary-phase method gave accurate results, except in the vicinity of z = v.

(18c)

For z > v, the Bessel function can be approximated as the superposition of the two stat ionary-point contributions , JvI ( z)and Jz (z). However, for z < v, the deformed integration contouronly passes through the second stationary point on the positive realaxis and therefore the Bessel function can be approximated as thecontribution of the second stationary point, Ju2 ( z) .

The Type-ill, first-kind Bessel function representation oforder 3 J3 (z) , computed via the stationary-phase method, isplotted in Figure 12 and compared with the MATLAB built-inbes s e 1 j function. The stationary-phase method gave accurateresults, except in the vicinity of z = v .

(15a)

(15b)1f t = -..21Z

q (t)= z sinh (t)- ot ,

3.1.4 Type IVFor the fourth Bessel integral representation given in Equa

tion (4), the magnitude and phase functions become

The stationarypoints are

(16a)

(16b) q(t)= -izt + io arcsin (t ) , (19a)The location,of these stationary points is shown in Figure 3c. Forz > v, ts l and ts2 are pure imaginary: ts l is on the negative imagi-nary axis, and ts2 is on the positive imaginary axis. They arelocated symmetrically along the imaginary t axis. If z increases, tsIand ts2 move away from each other. For z < v, tsl and ts2 arereal: tsI is negative, and ts2 is positive. They are locatedsymmetrically along the real t axis. If z decreases (moves awayfrom v , tsl and ts2 move away from each other. For z = v, ts land ts2 are both equal to zero. They are no longer isolated: theycoincide.

1f(t)= C-;21 \}I-t 2The stationarypoints are

(19b)

(20a)

(20b)

as


as

The locations of these stationarypoints are shown in Figure 3d. Forz > v, tsl and ts2 are real: ts l is negative, and ts2 is positive.They are located symmetrically along the real t axis. If z increases,ts1 and ts2 move away from each other. tsl is in the interval(-1,0), and ts2 is in the interval 0,1). For z < v, tsl and ta arepure imaginary: tsl is on the negative imaginary axis, and ts2 is onthe positive imaginary axis. They are located symmetrically alongthe imaginary t axis. If z decreases (moves away from v , t sI andts2 move away from each other. For z = v, tsl and ts2 are bothequal to zero. They are no longer isolated: they coincide.

Contrary to the Type- I Bessel representation, the stationarypoints are dependent on z and v. They are isolated for z values outside the vicinity of v. However, for z values in the vicinity of v,the stationary points are no longer isolated, and the stationaryphase method becomes invalid.

The contribution of the first stationary point can be computed

(17a)

(18a)

(17b)(17c)

q (tsl ) = z sinh(t sl ) ,

The contribution of the first stationary point can be computed

Contrary to the Type- I Bessel representation, the stationarypoints are dependent of z and v. They are isolated for z values outside the vicinity of v. However, for z values in the vicinity of v,the stationary points are no longer isolated, and the stationaryphase method becomes invalid.

q (ts2 ) =z sinh (ts2 ) , (18b) q (tsl ) =-iztsl + iu arcsin (tsl ) , (21a)

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O'US2O.,

SOP I,.1

,

M

' . . . . . . . 0. . . . . . . ' .u . . . .

t Dio.et. . . . . . . . .

14.1

~rZ:n

0 .

o.

d r : O r d ~

rl :rl : riO

d.. r :ri S- - . tO . . . . . . . , .Ord... :ns r- ................ .( 0 ............

O.D0t Ot0

IQ 2D :IS 30 m.nc.l8 ..... Figure lOa. The front panel of BESSEL_GUI: The Type-IVBessel function of order 15 computed via the stationary-phasemethod, steepest-decent-path evaluation, and the MATLABbuilt-in besse l j function, plotted from z =0 to z =30.

Figure lOb. The front panel of BESSEL_GUI: The Type-IVBessel function of order 15 computed via the stationary-phasemethod, steepest-decent-path evaluation, and the MATLABbuilt-in besse l j function, plotted from z =14 to z =16 nearz=v .

ss

. .. ...,'L.UI t Irl . fTrl : ,......., ...... r .

..r. ri I . s...... ,_:r r . ..... .r 0 . . - . . . . . . . . .01osO.;;;

4000

2D :IS 30 os 2Sw .\A.LIJ .t.. .. .. .

IS.0

~ - .' .....o . . -P .

( Dio.et. . . . . . . . .

~ s - - _ ~ ~ __ ---; ;--_ _ - - - ; ~ _ _ - - , ; -__ - - , ; -_ _ ---d

rl ;rrrl ; rJ D

4elr: r i1Order .

Figure 11.The front panel ofBESSEL_GUI:The Type-I Besselfunction of order 3 computed via the stationary-phase method,steepest-decent-path evaluation, and the MATLAB built-inbesse l j function, plotted from z =0 to z =30.

Figure 12. The front panel of BESSEL_GUI: The Type-IllBessel function of order 3 computed via the stationary-phasemethod, steepest-decent-path evaluation, uniform asymptotics,and the MATLAB built-in besse l j function, plotted from z =0to z = 6.

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( ) ivtslq tsI = ~ / 3v z

(21b)

(21c)

(21d)

through the second stationarypoint, as can be seen in Figure 3. Thesteepest-decent path can be found analytically, and can be visualized graphically by plotting the amplitude of the integrand in threedimensions. Finding the steepest-decent path and evaluating theBessel integral along this path is explained below only for the second-type Bessel integral representation, since the calculations forall types are very similar.


( ) _ ivts2q ts2 ~ / 3v z

q( ts2 ) = -izts2 + iv arcsin(ts2 ) ' The amplitude of the integrand for the second-type Besselfunction integral representation is plotted in Figure 6a for z > v .As mentioned in the stationary-phase method, the stationary pointsare located symmetrically along the real t axis. For the positive realstationary point ts2 the local steepest-decent path makes an angleof nl4 with the real axis. For the negative real stationary point tslthe local steepest-decent path makes an angle of - n14 with thereal axis. The stationary points and the steepest-decent path areplotted in Figure 6c.

3.2.1 Graphical Visualizations of theSteepest-Decent Paths(22a)

(22b)

(22c)

(22d)2n v3 -izts2+iuarcsin(ts 2 ) Z e --ivts2 z3 2nvFor z > V , the Bessel function can be approximated as the superposition of the two stationary-point contribut ions, JuI ( z)and Ju2 ( z). However for z < v, the deformed integration contouronly passes through the second stationary point on the posit iveimaginary axis, and therefore the Bessel function can be approxi-mated as the contribution of the second stationary point, Ju2 ( z).

The Type-N, first-kind Bessel function representation oforder 15 J15 (z)), computed via the stationary-phase method, isplotted in Figure 10 and compared with the MATLAB built-inbesse l j function. The stationary-phase method gave accurateresults, except in the vicinityof z = v .

3.2 Steepest-Descent Path Method SDPMThe steepest-decent path passes through the stationary point

in the direction of the steepest decrease in the amplitude of theintegrand. The results of the stationary-phase method can beimproved by numerical integration of the approximated phasefunction along the steepest-decent path [3]:

The amplitude of the integrand for the second-type Besselfunction representation is plotted in Figure 6b for z < v. As mentioned in the stationary-phase method, the stationary points arelocated symmetrically along the imaginary t axis. For the positiveimaginary stationary point ts2 the local steepest-decent path isparallel to the real t axis. For the negative real stationary point tslthe local steepest-decent path is parallel to the imaginary t axis, butthe deformed integration contour passes only through the secondstationary point, and the first stationarypoint does not contribute tothe numerical integration. The stationary points and the steepestdecent path are plotted in Figure 6d.

3.2.2 Analytical Determination of theSteepest-Decent PathsThe steepest-decent paths can also be found analytically. By

replacing the phase function with the first three terms of its Taylorexpansion, the integrand can be written as f (t )eO.5q(ts )( t-ts )2 . Theexponent in the integrand can be written as

(25)where p is the amplitude and If/ is the phase function, given as

;Along the steepest-decent path, ep e should decay and be equal toe- p . The phase function If/ can therefore be found as (eif// = -1 )

where the phase function is replaced with its first three terms of theTaylor series near t = ts :

(24)

If/=arg [q (ts ) ] + 2arg(t- ts ) .

If/ = n .

(26)

(27)The steepest-decent paths for all four types of integral Bessel representations are shown in Figure 3. For Types IT, Ill, and N, ifz > v, the integration contour is so deformed that it passes throughboth stationary points in the direction of the steepest-decent path.However, for z < v, the deformed integration contour passes only

If If/ = n is set into Equation (26), one obtains(28)

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Table 2. Analytical expressions of the steepest-decent paths for Z > v and Z < u .Fo r Z > v

tsi is real and negative.tsl =-b, b>Oq (tsl) =izsin(tsl) =-ci, c > 0arg[q (ts ) ] = -if/2i) arg(t - ts) = if/2 + if/4 =3if /4Im (t)> 0 part of Cl in Figure 6c.ii) arg(t - ts ) =- i f/2 + if/4=- if/4Im (t)< 0 part of Cl in Figure 6c.ts2 is real and positive.ts2 = b, b > 0q (ts2) = izsin(ts2) = ci, c> 0arg[q (ts )J= if/2i) arg (t- ts ) = if/2 - if/4 = if/4Im t > 0 part of C2 in Figure 6c.ii) arg(t - ts) =-if/2 - if/4 =-3if/4Im (t)< 0 part of C2 in Figure 6c.

For Z 0q (ts i) = iz sin (ts i) = c, c > 0 ,arg[q (ts)J = 0i) arg(t-ts ) = i f / 2Im (t)> tsI part of Cl in Figure 6d.ii) arg(tts ) = -if/2Im t < tsI part of Clin Figure 6d(this contour does not contribute)ts2 is imaginary and positive.ts2 =bi, b > 0q (ts2)=izsin(ts2)=-c , c > 0arg[q (ts ) ] =ifi) arg (t - ts ) = if/2 - i f/2 =0Re(t)> 0 part of C2 in Figure 6d.ii) arg(t - ts ) =- i f/2 - if/2 =-i fRe(t)< 0 part of C2 in Figure 6d.

Having determined the steepest-decent path, the phase function can be replaced with qI(t) around the first stationary pointand with q2(t) around the second stationary point, where qI(t)and q2(t) are determined according to Equation (24) as

The stationary points, tsI and ts2 of the second-type Bessel integral were given in Equation (12). The steepest-decent pathsthrough these stat ionary points are found according to Equation (28), and are summarized in Table 2 for both of the casesz > v and z v. For z < v, theintegration is done only along the second path, C2 , shown in Fig-ure 6d. A MATLAB module to calculate the first-type Bessel Function via the stationary-phase method and the steepest-decent path isgiven in Table 3.

The Type-II, first-kind Bessel integral of order 4 J4 (z)),computed via steepest-decent-path evaluation, is plotted in Figure 7 and compared with the stationary-phase method and theMATLAB built-in besse l j function. As can be seen in the figure,numerically integration of the approximated phase function alongthe steepest-decent path improved the results obtained via the stationary-phase method in the vicinity of z = v.

Replacing these phase functions in Equation (23), the Bessel function can be determined as

. . ( ). izsin(tsd( )21 f -lzsm tsl +IUtsl+--- t- tslJu(z)=- e 2 dt2if Cl

. . ( ). izsin(ts2)( 21 f -lzsm ts2 +IU s2+- - - t- ts2+- e 2 dt , (30)2i f C2IEEEAntennasandPropagation Magazine 1 Vat.51,No.6, December 2009

3.3 Uniform Asymptotics UABoth the stationary-phase method and the steepest-decent

path results become invalid near the caustic [4], where the two stationary points are no longer isolated. In this case, the integral canbe evaluated by uniform asymptotics in terms of Airy functions [1]and their derivatives as

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where ao ,S, and A1,2 are defined as

******************

nu=2; k=l; dx=O.OI;fo r z=O.1 :0.1:20coefl =exp(i*z)/(2*pi*i*i (nu+1));coef2=exp(-i*z)/(2*pi*i*(-i) (nu+I)); integ l=0; integ2=0;

for a=-2:dx:2tl =a+i*(-0.5*a+I); f1=exp(-i*z*0.5*(tl-i) 2);t2=a+i*(0.5*a-I); f2=exp(i*z*0.5*(t2+i) 2);integl=integl+fl *dx*(1-0.5i);

integ2=integ2+f2*dx*(1+O.5i);endbesl(k)=-coefl *integl+coef2*integ2; % SDPbes2(k)=sqrt(2/(pi*z))*cos(z-pi/4-pi*nu/2); % SPMbes3(k)=besselj(nu,z); zz(k)=z; k=k+l; % MatLab

endplot(zz,besl, b. ,zz,bes2, r-- ,zz,bes3, k )legend( SDM , 'SPM','MatLab',0)% ****************** END

Table 3. A MATLAB module calculating the Type-I, first-kindBessel function via SPM and SDP evaluation.

(32a)

(32c)

(32b)

(32d)

A1,2 = h12f (ts12) '

Here, Ai s is the Airy Function, and Ai s is the derivative ofthe Airy Function (with respect to the argument). For the fourthtype Bessel, the amplitude function, phase function, and the sta-tionary points were f(t) = 2 n ~ 1 - t 2 ),q (t ) = -izt + iv arcsin (t ) , and tsl,2 = + ~ I - v 2/ z2 , respectively.

oJ ) 1 fe-iZSin(-R+iY)+iV(-R+iY)zdyv z = 21 00

The Type-IV, first-kind Bessel function of order 10 J10 (z)), computed via uniform asymptotics, is plotted in Figure 8 and comparedwith the MATLAB built-in besse l j function. The uniformasymptotics gave accurate results except at the point z = v.

0 R+ f e-izsin(R+iy)+ivR+iyid F + f e-izsin(t)+ivtdt .o -R

(34)

3.4 Direct Numerical Integration The first and third integrals can be written as a single integral asAll four types of Bessel-function integral representations canalso be evaluated via direct numerical integration. The integral

shown in Figure 2b was chosen as an example. The integrationcontour for the Type-IT, first-kind Bessel function shown in Figure 2b can be deformed into three straight lines as2r - - - - . .A.- - - , Rf e - izs in(t )+iv tdt + f e-izsin(t)+ivtdt

-R+ioo -R

00Ju ( z ) = i [e-izsin(1T+iy)+iu(lI'+iy) - e-izsin(-lI +iy)+iu(-1T+iy)}tr

2R+ f e-izsin(t)+ivtdt . (35)-R

By simplifying the above equation, one getsR+ioo+ f e-izsin(t)+ivtdt .

R

(33)

1+3Ju (z)= -sin(un) j -zsinh y -uydr +. - fcos[zSin(t)-ut}it.1 0 1 0

(36)The first and third integrals are combined by introducingt = -1 + i r for the first integral and t = 1+ iF for the third integral:

For integral order v = m= 0,1,2, the factor sin (V1 in front of thefirst integral becomes zero, so the Bessel function of integral orderreduces to

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The Type-IT, first-kind Bessel function of order 4 J4 (z)), computed via direct numerical integration, is plotted in Figure 9 andcompared with the MATLAB built-in be s s e l j function. The directnumerical integration also gave accurate results in the vicinity ofz = v , and exactly at the point z = v .

1 1rJu (z) = - Jos[zsin (t ) - vt }i t .7 0

(37) Figure 12 presents the graph of the Bessel function of orderv = 3 plotted in the range of 0 z 6 . The third-type Bessel integral was chosen and evaluated using the stationary-phase method,steepest-decent-path evaluation, and uniform asymptotics. The uniform-asymptotics method also gave accurate results near z = v .

5. Conclusions

4. The esse _GUI PackageThe MATLAB-based simulation package BESSEL_GUI, withthe front panel displayed in Figure 10, was prepared for theasymptotic evaluation of Bessel integrals. A pop-up menu allowsthe user to choose the type of the Bessel integral. For each type ofBessel integral, a figure containing the original integration contour,

the stationary points, tsl and ts2 and the steepest-decent pathappears at the top of the front panel. The corresponding formula ofthe Bessel integral is presented to the right of the figure. Jlist belowthe pop-up menu, the user may check one or more radio buttons inorder to choose which methods to use to calculate the Bessel integral. The user may also select from which zl start value to whichz2 end value the Bessel function is to be calculated. The computation step, t:1z, and the order of the Bessel function are also specified by the user at the upper left part. Once the user presses thePlot button, the Bessel function of user-defined order is calculated from the user-defined zl value to the user-defined z2 value,using the methods chosen, and plotted to the graph. The Besselfunction computed via the MATLAB built-in function is also plotted to the graph, to validate each method. The tool also allowsplotting the magnitude of the complex integrand of the Bessel integral in three dimensions, as in Figures 6a and 6b. The user thus canvisualize the locations of the stationary points, and also the steepest-decent paths.

The graph of the Bessel function of order v = 15 J15 ( z)),plotted from zl = 0 to Z2 = 50 , with a computation step of& = 0.1, is shown in Figure lOa. In this example, the fourth-typeBessel integral was chosen and evaluated using the stationaryphase method and steepest-decent-path evaluation. Both asymptotic methods gave accurate results, except in the vicinity ofz = v = 15. Figure lOb presents the same graph, only in theproblematic range 14 z :5; 16. It was obvious that numerical integration of the approximated phase function along the steepestdecent path improved the results obtained via the stationary-phasemethod in the vicinity of z = v = 15 .

The Bessel function of order v = 3 is plotted in Figure 11 forthe range of 0:5; z 30 . Here, the first-type Bessel integral waschosen, and the formula of the first-type Bessel integral is presented in the upper-right part. The original integration contour, Ct ;the stationary points, tsl and ts2 ; and the steepest-decent path areshown in the figure. The integral was evaluated using the stationary-phase method and steepest-decent-path evaluation. Both methods approximated the Bessel function only for z v .

Bessel functions and asymptotic evaluation of Bessel integrals have been investigated. High-frequency asymptotic methods,such as stationary phase, steepest-decent-path evaluation, and uniform asymptotics, were used to evaluate four different types ofBessel integrals. A user-friendly MATLAB-based virtual tool wasalso introduced. The tool enables the user to plot the Bessel function in a user-specified range by evaluating one of the four different types of Bessel integrals, using one or more of the above-mentioned asymptotic approaches, or using direct numerical integration. It compares the results with the MATLAB built-in be s s e l jfunctions. The tool also enables visualization of the locations ofthe saddle points and also the steepest-decent paths by plotting themagnitude of the complex integrand of the Bessel integral in threedimensions. Some illustrative examples were presented to showthe attraction of the package. The tool may be used in both undergraduate and graduate lectures.

6. References1. M. Abramowitz and I. Stegun, Handbook of MathematicalFunctions, New York, Dover Publications Inc., 1965.2. N. N. Lebedev, Special Functions & Their Applications, NewYork, Dover Publications Inc., 1972.

3. L. B. Felsen and N. Marcuvitz, Radiation and Scattering ofWaves,Upper Saddle River, NJ, Prentice-Hall Inc., 1973.4. L. Sevgi, S. Paker, and E. Topuz, Intrinsic Mode (IM) Formalism and Its Asymptotical Evaluations in 3D Non-HomogeneousEnvironments, AEU, Int. J. of Electronics and Communication,50, 3, May 1996, pp. 201-207.5. F. Akleman and L. Sevgi, Analytical and Numerical Investigations of Ring Resonators, ELEKTRIK, Turkish Journal ofElectrical Engineering and Computer Sciences, 16, 1, January 2008, pp.87-94.6. S. A. Bechteler and L. Sevgi, Millimeter Waveband SemiSymmetrical Groove GuideResonator, IEEE Microwave Magazine, 5, 3, September 2004, pp. 51-60.7. L. Sevgi, F. Akleman, and L. B. Felsen, Visualizations of WaveDynamics in a Wedge-Waveguide with Non-Penetrable Boundaries: Normal, Adiabatic, and Intrinsic-Mode Representations,IEEE Antennas and Propagation Magazine, 49, 3, June 2007, pp.76-94.00

A Tutorial on Bessel Functions

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