Comparative Study on the Computational Modeling and Application of the Classes of A, B, C Numbers

Mariana Nikolova Georgieva-Grosse 1 Georgi Nikolov Georgiev

2

1 Consulting in Physics and Computer Sciences, Meterstrasse 4, D-70839 Gerlingen, Germany, e-mail: mngeorgieva@yahoo.de. 2 Faculty of Mathematics and Informatics, University of Veliko Tirnovo “St. St. Cyril and Methodius”, BG-5000 Veliko Tirnovo, Bulgaria, e-mail: g.georgiev@uni-vt.bg.

Abstract − The classes 1A , 1B , 1C and 2A , 2B , 2C of positive real numbers are defined by means of the positive purely imaginary zeros of a complex Kummer confluent hypergeometric function and of a complex function, composed of two Kummer and two Tricomi confluent ones, resp. of specially selected parameters. Based on numerical examples, methods for computational modeling of these quantities, are presented. The comparison between them reveals that finding the values of elements of the second group is much more complicated. Their slighter dependence on certain par-ameters, however, allows to get better results in employing a definite scheme for approximate calculation of the differential phase shift, raised by the coaxial ferrite waveguide with azi-muthal magnetization for the normal nTE0 modes than when the same procedure, harnessing the first set of numbers, is applied to a circular configuration, filled in by the medium mentioned.

1 INTRODUCTION

The A , B and C numbers appeared in the theory of the azimuthally magnetized circular and coaxial ferrite waveguides, propagating normal nTE0 modes [1-5], based on the complex confluent hypergeometric func-tions [6], as a means for computation of the differential phase shift, afforded by these structures [1]. The elaborate way in which they have been advanced [2] and the insufficient yet information for them [1-5] however, hampers their usage in practice.

Here, a classification of these quantities is made. The classes 1A , 1B , 1C and 2A , 2B , 2C are regarded. A new definition of each of them is given in terms of the numbers −1A , +1A , −1B , +1B , −1C , +1C and −2A , +2A ,

−2B , +2B , −2C , +2C , resp., specified in parametric form. The first (second) group, labelled by the subscript ±1 , resp. 1 ( ±2 , resp. 2) is initiated by the positive purely imaginary zeros of a complex Kummer confluent hypergeometric function (of a complex function, being a combination of Kummer and Tricomi ones) [6] of appropriately picked out parameters. The numbers are modelled, employing the method of successive approximations. The techniques, used in the two cases, are juxtaposed. Graphs visualize the behaviour of quantities 2A , 2B , 2C . A comparison is made on the application of both kinds of numbers in the theory of waveguides.

2 1A , 1B , 1C AND 2A , 2B , 2C NUMBERS

Definition 1: The positive real numbers ±iA , ±iB and ±iC ( 2,1=i ), depending on the set of quantities

( )0,,, rnc α if 1=i and ( )0,,,, rnc αρ in case 2=i , are determined in parametric form by the equations: αβ /±± =iA , (1) α/±± = ii KB , (2) 2/α±± = ii KC , (3) ( )±±= βα/0 iKr , (4) with ( ) ( )( )[ ]{ } 2/122 2/1/1 ±± +−= kααβ and α and

±k as parameters. (The subscripts “+” and “–” answer to 0>+k and 0<−k , resp.)

i) On the understanding that 1=i , it is valid: ( )±±± = kAA ,11 α , ( )±±± = kBB ,11 α , ( )±±± = kCC ,11 α ,

( )±= kncrr ,,,00 α , ( )±±± = kncKK ,,11 , ( ) =±± kncK ,,1 ( )c

nkk ,±± ζ , ( ) ( )cnkakncM ,1 ,,

±±±± = ζ , ( )cnk ,ζ – n th positive

purely imaginary zero in x of the function ( ) =xcaF ;,1 ( )xca ;,Φ and the pair { }0, rα of positive real numbers

1<α and ( ) 2/,00cnr ζ> subordinates to the condition:

( ) ( ) ααζ /,12/ 1

20,0 ncLrc

n <−< (5) in which ( )ncL ,1 are positive real numbers, brought in by the relation: ( ) ( ) ( )−−−∞→−−−∞→ −−

== kncMkncKncLkk

,,lim,,lim, 111 . (6)

ii) Provided 2=i , it is fulfilled: ( )±±± = kAA ,22 α , ( )±±± = kBB ,,22 αρ , =±2C

( )±± kC ,,2 αρ , ( )±= kncrr ,,,,00 αρ , =±2K ( )±± kncK ,,,2 ρ , ( ) ( ) ( )ρχρ c

nkkkncK ,2 ,,,±±±± = ,

( ) ( ) ( )ρχρ cnkakncM ,1 ,,,

±±±± = , ( ) ( )ρχ cnk , – n th positive

purely imaginary zero in x of the function ( ) ( ) ( ) ( ) ( )xcaxcaxcaxcaxcaF ;,;,;,;,,;,2 ΨΦΨΦ −= ρρρ

and the triad { }0,, rαρ of positive real numbers 1<α , ( ) ( ) 2/,00 ρχ c

nr > , 10 << ρ meets the criterion:

978-1-61284-978-2/11/$26.00 ©2011 IEEE

660

( ) ( ) ( ) αραρχ /,,12/ 22

0,0 ncLrcn <−< (7)

in which ( )ncL ,,2 ρ are positive real numbers, introduced, as follows:

( ) ( ) ( ).,,,lim,,,lim,, 222 −−−∞→−−−∞→ −−== kncMkncKncL

kkρρρ

(8) ( )xca ;,Φ and ( )xca ;,Ψ are the Kummer and Tricomi

confluent hypergeometric functions [6], with =a jkc −2/ – complex, 3=c , jzx = – positive purely im-

aginary, k , z – real, +∞<<−∞ k , 0>z , ( ,...3,2,1=n ). In case 3=c , 1=n and 1.0=ρ , ( ) =ncL ,1 6.59365

41068 [3] and ( ) =ncL ,,2 ρ 7.65009. Definition 2: 2,1A , 2,1B and 2,1C numbers are

called the real quantities: ++−− −= 2,12,12,1 AAA , (9) ++−− −= 2,12,12,1 BBB , (10) ++−− −= 2,12,12,1 CCC (11) where −− 2,1A , ++ 2,1A , −− 2,1B , ++ 2,1B , −− 2,1C , ++ 2,1C are specified for the same α and for the values of

+k , −k , ++ 2,1K and −− 2,1K , subject to the requirement +−++−− = ββ // 2,12,1 KK which conforms to the condi-

tion −+ = 00 rr . ( −− 2,1A , ++ 2,1A , −− 2,1B , ++ 2,1B , −− 2,1C , ++ 2,1C , +k , −k , ++21K , −− 2,1K , 0r and α are

considered in the sense of Definition 1.)

3 COMPUTATIONAL MODELING OF THE 1A , 1B , 1C NUMBERS

The modeling of the 1A , 1B , 1C numbers is illus-trated by the following numerical example. Let

=chα 0.2, 50 =chr and 1=chn is a triad of quantities, satisfying the criterion (5). Besides, let =−

chk –0.1 be an arbitrarily picked out negative value of the par-ameter k . For it ( ) =

−

compcnk chch ,ζ 7.26070 46121 and

( ) =−−chchcomp kncK ,,1 0.72607 04612. Introducing the

numerical equivalents of chα , chk− and ( )chchcomp kncK −− ,,1 in the expressions (1)-(4) gives: =−

compr0 5.23996 22029, =−compA1 3.46410 16151,

=−compB1 3.63035 23061, =−

compC1 18.15176 15303. As chcomp rr 00 ≠− , the procedure should go on. Taking now =−

chk –0.2, again arbitrarily, yields: ( ) =−

compcnk chch ,ζ

6.87510 73455, ( ) =−−chchcomp kncK ,,1 1.37502 14691,

=−compr0 7.84510 70383, =−

compA1 4.38178 04600, =−

compB1 6.87510 73455, =−compC1 34.37553 67273.

Like before, chcomp rr 00 ≠− . Choosing =−chk –0.05 re-

sults in: ( ) =−

compcnk chch ,ζ 7.46000 34112, ( ) =−−

chchcomp kncK ,,1 0.37300 01706, =−

compr0 4.25626 26535, =−compA1

2.19089 02300, =−compB1 1.86500 08528, =−

compC1

9.32500 42640. If ±iD typifies any of the numbers ±iA , ±iB or ±iC , in the above three cases it holds:

≡−compD1 ( )compchchcomp rncD −− 01 ,,, α . At this step the

arbitrary variation of k terminates, because an interval =Δ −

)0(0compr [ ])0(

,0)0(

,0 , compright

compleft rr −− ( =−

)0(,0

compleftr

4.38178 04600, =−)0(

,0comp

rightr 5.23996 22029) is fixed, containing the originally selected numerical equival-ent of 0r . Thus, an interval (intervals) (zeroth ap-proximations) for chk− (for compA −1 , compB −1 and compC −1 ) is (are) determined, involving the values, looked for. In particular =−

)0(,

chleftk –0.1, =−

)0(,

chrightk –0.05. Next, the

interval for chk− is divided in 100 parts. As first approximations for the quantities studied are accept-ed those, pertinent to the limits of the smallest interval of values for 0r , comprising chr0 : =−

)1(,

chleftk –0.0895,

=−

)1()(,,)1(

,

compcleftnk chch

leftζ 7.30220 76841, ( ) =−−

)1(,

)1(,1 ,, ch

leftchcomp

left kncK 0.65354 75877, =−

)1(,0

comprightr 5.00090 30956, =−

)1(,1

comprightA

3.26714 78292, =−)1(

,1comp

rightB 3.26773 79386, =−)1(

,1comp

rightC 16.33868 96931, ( ));( )1(

,0)1(

,1)1(

,1 ,,, compright

chchcompright

compright rncDD −−− = α

=−)1(

,ch

rightk –0.0890, =−

)1()(,,)1(

,

compcrightnk chch

rightζ 7.30418 87065,

( ) =−−)1(

,)1(

,1 ,, chright

chcompright kncK 0.65007 27949, =−

)1(,0

compleftr

4.98985 00240, =−)1(

,1comp

leftA 3.25697 56193, =−)1(

,1comp

leftB 3.25036 39744, =−

)1(,1

compleftC 16.25181 98720,

( =−)1(

,1comp

leftD ( ))1(,0

)1(,1 ,,, comp

leftchchcomp

left rncD −−− α ). Then, the process is reiterated. (The left ends of certain of the intervals might correspond to the right ends of other ones.) The upshot (the sixth approximation) is:

=−)6(compk –0.08945 92019, =

−

)6()(,)6(

compcnk chcompζ 7.30236

93121, ( ) =−−)6()6(

1 ,, compchcomp kncK 0.65326 41306, =−

)6(0compr 5.00000 00000, =−

)6(1compA 3.26632 06529,

=−)6(

1compB 3.26632 06529, =−

)6(1compC 16.33160 32644,

( =−)6(

1compD ( ))6(

0)6(

1 ,, compchchcomp rncD −− α ). Similarly, with =+

)0(,

chleftk 0.05 and =+

)0(,

chrightk 0.1 as ends of the

initial interval, for the sixth approximation it is obtained: =+

)6(compk 0.07171 90295, =+

)6()(,)6(

compcnk chcompζ

7.96197 43928, ( ) =++)6()6(

1 ,, compchcomp kncK 0.57102 50762, =+

)6(0compr 5.00000 00000, =+

)6(1compA 2.85512

53811, =+)6(

1compB 2.85512 53811, =+

)6(1compC 14.27562

69056, ( =+)6(

1compD ( ))6(

0)6(

1 ,,, compchchcomp rncD ++ α ). At last, the values of numbers sought, counted with an accuracy 1010−=chε are: =compA1 0.41119 52718,

=compB1 0.41119 52718, =compC1 2.05597 63588 ( =compD1 ( )chchchcomp rncD 01 ,,, α ). (The superscripts “ ch ” (“ comp ”) label the symbols, standing for the parameters chosen (for the quantities computed). Moreover, everywhere the digits in a given result, being identical with these in the final one, are marked off by bold face type.) The dependence of 1A , 1B ,

1C on α and 0r is shown for 1=n in Figs. 1-3 [3].

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Figure 1: 2A numbers vs. α in the interval ⟩÷⟨ 0.10

with 0r as parameter for 1.0=ρ and 1=n .

(a)

Figure 3a: 2C numbers in the interval ⟩÷⟨ 5.20 vs. 0r in the interval ⟩÷⟨ 104 with α as par-ameter for 1.0=ρ and 1=n ;

Figure 2: 2B numbers vs. 0r in the interval ⟩÷⟨ 104

with α as parameter for 1.0=ρ and 1=n .

(b)

Figure 3b: 2C numbers in the interval ⟩÷⟨ 975.191.1 vs. 0r in the interval ⟩÷⟨ 104 with α as par-ameter for 1.0=ρ and 1=n .

4 COMPUTATIONAL MODELING OF THE 2A , 2B , 2C NUMBERS

The idea of the modeling is the same as in case of the 1A , 1B , 1C numbers. Unlike before, however, now a

set of four quantities: ρ , α , 0r and n is taken into account. Let 1.0=chρ , =chα 0.2, =chr0 5 and =chn 1. As can easily be checked, they satisfy the condi-tion (7). It may be established as well that the intervals with ends =−

)0(,

chleftk –0.1, =−

)0(,

chrightk –0.05 and

=+)0(

,ch

leftk 0.05 and =+)0(

,ch

rightk 0.1 are suitable for per-forming the procedure, described in Section 3. The final results for 1010−=chε are: =−

)6(compk ,348190.08218− ( ) =

+

chcompcnk chcomp ρχ )6()(

,)6( 7.56958 19612, ( ) =−−

)6()6(2 ,,, compchchcomp kncK ρ 0.62210 47254,

=−)6(

0compr 5.00000 00000, =−

)6(2compA 3.11052 36268,

=−)6(

2compB 3.11052 36268, =−

)6(2compC 15.55261 81342,

( ( ))6(0

)6(2

)6(2 ,,,, compchchchcompcomp rncDD −−− = αρ ), resp.;

=+)6(compk 0.06689 58481, ( ) =

+

chcompcnk chcomp ρχ )6()(

,)6( 8.143770

662

24965, ( ) =++)6()6(

2 ,,, compchchcomp kncK ρ 0.54478 44172, =+

)6(0compr 5.00000 00000, =+

)6(2compA 2.72392 20859,

=+)6(

2compB 2.72392 20859, =+

)6(2compC 13.61961 04297,

( ( ))6(0

)6(2

)6(2 ,,,, compchchchcompcomp rncDD +++ = αρ ); and

=compA2 0.38660 15409, =compB2 0.38660 15409, =compC2 1.93300 77046, ( ( )chchchchcompcomp rncDD 022 ,,,, αρ= ).

As seen, for the same chα , chr0 and chn , 21 AA > , 21 BB > and 21 CC > . The number of necessary

iterations is specified by the choice of initial interval, by chρ and chε . To find the values of 2A , 2B , 2C is much more complex than of 1A , 1B , 1C , since: i) the numbers depend on four parameters, instead of three ( 3=c is a constant); ii) the zeros of a more complicated function, involving both the Kummer and Tricomi ones, are used; iii) the logarithmic representation of the Tricomi function is employed. The effect of α and 0r on 2A , 2B , 2C is portrayed in Figs. 1-3 for =ρ 0.1 and 1=n . The dotted lines in all pictures describe the case 02 =+A , 02 =+B ,

02 =+C . The co-ordinates of points (circles) in them are summed up through the left-hand part of inequality (7) with the sign “ = ” instead of “ < ” in it. The comparison with Figs. 1-3 [3] indicates that iA ( iB ) is almost constant with respect to α ( 0r )and

iC practically is not influenced by both α and 0r . This tendency is more pronounced for 2A , 2B , 2C (especially when ρ gets larger) than for 1A , 1B , 1C .

5 COMPARATIVE STUDY ON THE

APPLICATION OF THE A, B, C NUMBERS

If α is the modulus of the off-diagonal element of the Polder permeability tensor, 0r and ±β are the normalized in a special way towards the frequency guide radius and phase constant for positive (counter-clockwise) and negative (clockwise) magnetization and ρ is the central conductor to guide radius ratio, then the differential phase shift +− −=Δ βββ , produced by the azimuthally magnetized circular and coaxial ferrite waveguides that support normal nTE0 mode, could be reckoned up by the formula [2,3,5]: CAB /=Δβ (12) in which the symbols A , B , C stand for 1A , 1B , 1C ( 2A , 2B , 2C ) in case the first (second) structure is treated. It may be shown also that the relations (5) and (7) give the conditions for phase shifter operation of the two configurations. As evident, the computation of numbers for each set of parameters is a complicated task. However, taking into account their slight depend-ence on α and 0r , it is possible to develop an approximate approach for finding A , B , C ( βΔ ) for any set of α and 0r (for fixed ρ ), if their values for one such set are known. In view of what has been said above, the results for larger ρ would be better.

6 CONCLUSION

A comparative investigation is carried out on the schemes for computational modeling and application of the classes of 1A , 1B , 1C and 2A , 2B , 2C numbers – positive real numbers, connected with certain zeros of the complex Kummer function and of a combination of complex Kummer and Tricomi functions, resp. It is shown that these quantities could be used to cast up the differential phase shift, provided by the circular and coaxial waveguides, containing azimuthally mag-netized ferrite for the normal nTE0 modes.

Acknowledgments

We express our gratitude to our mother Trifonka Romanova Popnikolova and to our late father Nikola Georgiev Popnikolov for their self-denial and for their tremendous efforts to support all our undertakings.

References

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[2] M.N. Georgieva-Grosse, and G.N. Georgiev, “The A, B, C numbers and their application in the theory of waveguides,” in Proc. 26th Progr. In Electro-magn. Res. Symp. PIERS 2009, Moscow, Russia, p. 612, in Abstracts, pp. 1043–1047, Aug. 18–21, 2009.

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[4] M.N. Georgieva-Grosse, and G.N. Georgiev, “On the class of 1A numbers: Definition, numerical modeling, domain of existence and application,” in Proc. XXX URSI General Assembly, in CDROM, Istanbul, Turkey, article ID BP1.47, 4 pages, Aug. 13–20, 2011.

[5] M.N. Georgieva-Grosse, and G.N. Georgiev, “Ad-vances in the theory of A , B , C numbers and its application to waveguide propagation,” in Proc. 30th Progr. In Electromagn. Res. Symp. PIERS 2011, Suzhou, China, Sept. 12–16, 2011, (in print).

[6] F.G. Tricomi, Fonctions Hypergéométrique Con-fluentes, Paris, France: Gauthier-Villars, 1960.

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