On the L4 numbers and their application
The F 4 set of complex functions: F a 4 =F a 4 (a,c;x,ε a ,ρ,α), F̅ 4 = F̅ 4 (a,c;x,̅μ,ρ,α), F̃ 4 = F̃ 4 (a,c;x,ε̃,ρ,α), equation and ̃F̃ 4lim = F̃ 4lim (a,c;x,ε̃,ρ,α) is defined by means of the complex Kummer and Tricomi confluent hypergeometric functions Φ(a,c;x) and Ψ(a,c;x), and the real ordinar...
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| description | The F 4 set of complex functions: F a 4 =F a 4 (a,c;x,ε a ,ρ,α), F̅ 4 = F̅ 4 (a,c;x,̅μ,ρ,α), F̃ 4 = F̃ 4 (a,c;x,ε̃,ρ,α), equation and ̃F̃ 4lim = F̃ 4lim (a,c;x,ε̃,ρ,α) is defined by means of the complex Kummer and Tricomi confluent hypergeometric functions Φ(a,c;x) and Ψ(a,c;x), and the real ordinary and modified Bessel ones, J v (y) and I v (u), (a=c/2-jk-complex; c=3; x=jz; k, z, equation,ε a , ε̅, ρ, α y, u - real; -∞ ; 1,ε̅ = 1, 0 |
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N.</creatorcontrib><description><![CDATA[The F 4 set of complex functions: F a 4 =F a 4 (a,c;x,ε a ,ρ,α), F̅ 4 = F̅ 4 (a,c;x,̅μ,ρ,α), F̃ 4 = F̃ 4 (a,c;x,ε̃,ρ,α), equation and ̃F̃ 4lim = F̃ 4lim (a,c;x,ε̃,ρ,α) is defined by means of the complex Kummer and Tricomi confluent hypergeometric functions Φ(a,c;x) and Ψ(a,c;x), and the real ordinary and modified Bessel ones, J v (y) and I v (u), (a=c/2-jk-complex; c=3; x=jz; k, z, equation,ε a , ε̅, ρ, α y, u - real; -∞ <; k <; +∞ in case of F̅ 4 and F̅ 4 ; and |k| <; |K lim |, |k| >; |K lim |, and |k| = |K lim | = (|α|/2)√ε̃ /[(1-α 2 )-ε̃] in case of F̃ 4 ,F̃ 4 and F̃ 4lim , resp.; z >; 0; ε a >; 1,ε̅ = 1, 0 <; ε̃ <; 1; 0 <; ρ <; 1, if F a 4 and F̅ 4 ; |α| >; |α lim |, if̃ F̃ 4 ; |α| <; |α lim |, if F̃ 4 and |α| = |α lim |, if F̃ 4lim are (is) concerned, (|α lim |= √1 - ε̃); α - <; 0, α + >; 0; y >; 0, u >; 0; v=0.1). The infinite sequences of positive real numbers K 4± (c,ε,ρ,α ± ,n,k ± =|k ± η (c) k± ,n(ε,ρ,α ± ) and M 4± (c,ε,ρ,α ± ,n,k ± )= |a ± |η (c) k± ,n(ε,ρ,α ± ) are constructed in which η k± , (c) n(ε,ρ,α ± ) are the n th positive purely imaginary zeros of the F 4 functions in x, n=1,2,3, sgn k= sgn α and the parameters acquire the values pointed out. The class of L 4 numbers, involving the subclasses L a 4± = L a 4± (c,ε a ,ρ,α ± ,n), L̅ 4± = L̅ 4± (c,̅ε,ρ,α ± ,n), L̃ 4± = L̃ 4± (c,̅ε,ρ,α ± ,n), L̃ 4± = L̃ 4± (c,̅ε,ρ,α ± ,n) and L̃ 4lim± = L̃ 4lim± (a,c;x,ε̃,ρ,α) is advanced. Each of its elements equals the common limits of the sequences mentioned, put together of the zeros of the respective F 4 function, attained, provided k + → +∞ or K - → -∞ (the subscripts "+", "-" correspond to the relevant sign of k). Some values of the quantities L 4 are computed and presented in a tabular form. The significance of the numbers considered for the theory of waveguides is debated.]]></description><identifier>ISSN: 2165-3585</identifier><identifier>ISBN: 1457708973</identifier><identifier>ISBN: 9781457708978</identifier><identifier>EISSN: 2165-3593</identifier><identifier>EISBN: 9789660259041</identifier><identifier>EISBN: 9660259042</identifier><language>eng</language><publisher>IEEE</publisher><subject>Antenna arrays ; Bars ; Equations ; Mathematical model ; Phase shifters ; Physics ; Waveguide theory</subject><ispartof>2011 XVIth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED), 2011, p.228-236</ispartof><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://ieeexplore.ieee.org/document/6081806$$EHTML$$P50$$Gieee$$H</linktohtml><link.rule.ids>309,310,778,782,787,788,2054,54903</link.rule.ids><linktorsrc>$$Uhttps://ieeexplore.ieee.org/document/6081806$$EView_record_in_IEEE$$FView_record_in_$$GIEEE</linktorsrc></links><search><creatorcontrib>Georgiev, G. N.</creatorcontrib><creatorcontrib>Georgieva-Grosse, M. N.</creatorcontrib><title>On the L4 numbers and their application</title><title>2011 XVIth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED)</title><addtitle>DIPED</addtitle><description><![CDATA[The F 4 set of complex functions: F a 4 =F a 4 (a,c;x,ε a ,ρ,α), F̅ 4 = F̅ 4 (a,c;x,̅μ,ρ,α), F̃ 4 = F̃ 4 (a,c;x,ε̃,ρ,α), equation and ̃F̃ 4lim = F̃ 4lim (a,c;x,ε̃,ρ,α) is defined by means of the complex Kummer and Tricomi confluent hypergeometric functions Φ(a,c;x) and Ψ(a,c;x), and the real ordinary and modified Bessel ones, J v (y) and I v (u), (a=c/2-jk-complex; c=3; x=jz; k, z, equation,ε a , ε̅, ρ, α y, u - real; -∞ <; k <; +∞ in case of F̅ 4 and F̅ 4 ; and |k| <; |K lim |, |k| >; |K lim |, and |k| = |K lim | = (|α|/2)√ε̃ /[(1-α 2 )-ε̃] in case of F̃ 4 ,F̃ 4 and F̃ 4lim , resp.; z >; 0; ε a >; 1,ε̅ = 1, 0 <; ε̃ <; 1; 0 <; ρ <; 1, if F a 4 and F̅ 4 ; |α| >; |α lim |, if̃ F̃ 4 ; |α| <; |α lim |, if F̃ 4 and |α| = |α lim |, if F̃ 4lim are (is) concerned, (|α lim |= √1 - ε̃); α - <; 0, α + >; 0; y >; 0, u >; 0; v=0.1). The infinite sequences of positive real numbers K 4± (c,ε,ρ,α ± ,n,k ± =|k ± η (c) k± ,n(ε,ρ,α ± ) and M 4± (c,ε,ρ,α ± ,n,k ± )= |a ± |η (c) k± ,n(ε,ρ,α ± ) are constructed in which η k± , (c) n(ε,ρ,α ± ) are the n th positive purely imaginary zeros of the F 4 functions in x, n=1,2,3, sgn k= sgn α and the parameters acquire the values pointed out. The class of L 4 numbers, involving the subclasses L a 4± = L a 4± (c,ε a ,ρ,α ± ,n), L̅ 4± = L̅ 4± (c,̅ε,ρ,α ± ,n), L̃ 4± = L̃ 4± (c,̅ε,ρ,α ± ,n), L̃ 4± = L̃ 4± (c,̅ε,ρ,α ± ,n) and L̃ 4lim± = L̃ 4lim± (a,c;x,ε̃,ρ,α) is advanced. Each of its elements equals the common limits of the sequences mentioned, put together of the zeros of the respective F 4 function, attained, provided k + → +∞ or K - → -∞ (the subscripts "+", "-" correspond to the relevant sign of k). Some values of the quantities L 4 are computed and presented in a tabular form. The significance of the numbers considered for the theory of waveguides is debated.]]></description><subject>Antenna arrays</subject><subject>Bars</subject><subject>Equations</subject><subject>Mathematical model</subject><subject>Phase shifters</subject><subject>Physics</subject><subject>Waveguide theory</subject><issn>2165-3585</issn><issn>2165-3593</issn><isbn>1457708973</isbn><isbn>9781457708978</isbn><isbn>9789660259041</isbn><isbn>9660259042</isbn><fulltext>true</fulltext><rsrctype>conference_proceeding</rsrctype><creationdate>2011</creationdate><recordtype>conference_proceeding</recordtype><sourceid>6IE</sourceid><sourceid>RIE</sourceid><recordid>eNp9ib0KwjAYAD__wKp9ApdsToWkSb4ksygOQhf3EjVipI0lqYNvr4I4esvB3QByo7RBpKU0VLAhZCVDWXBp-AhmTEilqDaKj39DyynkKd3oG0TDOMtgVQXSXx3ZCxIe7dHFRGw4f5KPxHZd40-29_ewgMnFNsnlX89hud0c1rvCO-fqLvrWxmeNVDNNkf-_L4K0L-s</recordid><startdate>201109</startdate><enddate>201109</enddate><creator>Georgiev, G. N.</creator><creator>Georgieva-Grosse, M. N.</creator><general>IEEE</general><scope>6IE</scope><scope>6IL</scope><scope>CBEJK</scope><scope>RIE</scope><scope>RIL</scope></search><sort><creationdate>201109</creationdate><title>On the L4 numbers and their application</title><author>Georgiev, G. N. ; Georgieva-Grosse, M. N.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-ieee_primary_60818063</frbrgroupid><rsrctype>conference_proceedings</rsrctype><prefilter>conference_proceedings</prefilter><language>eng</language><creationdate>2011</creationdate><topic>Antenna arrays</topic><topic>Bars</topic><topic>Equations</topic><topic>Mathematical model</topic><topic>Phase shifters</topic><topic>Physics</topic><topic>Waveguide theory</topic><toplevel>online_resources</toplevel><creatorcontrib>Georgiev, G. N.</creatorcontrib><creatorcontrib>Georgieva-Grosse, M. N.</creatorcontrib><collection>IEEE Electronic Library (IEL) Conference Proceedings</collection><collection>IEEE Proceedings Order Plan All Online (POP All Online) 1998-present by volume</collection><collection>IEEE Xplore All Conference Proceedings</collection><collection>IEEE Electronic Library (IEL)</collection><collection>IEEE Proceedings Order Plans (POP All) 1998-Present</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Georgiev, G. N.</au><au>Georgieva-Grosse, M. N.</au><format>book</format><genre>proceeding</genre><ristype>CONF</ristype><atitle>On the L4 numbers and their application</atitle><btitle>2011 XVIth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED)</btitle><stitle>DIPED</stitle><date>2011-09</date><risdate>2011</risdate><spage>228</spage><epage>236</epage><pages>228-236</pages><issn>2165-3585</issn><eissn>2165-3593</eissn><isbn>1457708973</isbn><isbn>9781457708978</isbn><eisbn>9789660259041</eisbn><eisbn>9660259042</eisbn><abstract><![CDATA[The F 4 set of complex functions: F a 4 =F a 4 (a,c;x,ε a ,ρ,α), F̅ 4 = F̅ 4 (a,c;x,̅μ,ρ,α), F̃ 4 = F̃ 4 (a,c;x,ε̃,ρ,α), equation and ̃F̃ 4lim = F̃ 4lim (a,c;x,ε̃,ρ,α) is defined by means of the complex Kummer and Tricomi confluent hypergeometric functions Φ(a,c;x) and Ψ(a,c;x), and the real ordinary and modified Bessel ones, J v (y) and I v (u), (a=c/2-jk-complex; c=3; x=jz; k, z, equation,ε a , ε̅, ρ, α y, u - real; -∞ <; k <; +∞ in case of F̅ 4 and F̅ 4 ; and |k| <; |K lim |, |k| >; |K lim |, and |k| = |K lim | = (|α|/2)√ε̃ /[(1-α 2 )-ε̃] in case of F̃ 4 ,F̃ 4 and F̃ 4lim , resp.; z >; 0; ε a >; 1,ε̅ = 1, 0 <; ε̃ <; 1; 0 <; ρ <; 1, if F a 4 and F̅ 4 ; |α| >; |α lim |, if̃ F̃ 4 ; |α| <; |α lim |, if F̃ 4 and |α| = |α lim |, if F̃ 4lim are (is) concerned, (|α lim |= √1 - ε̃); α - <; 0, α + >; 0; y >; 0, u >; 0; v=0.1). The infinite sequences of positive real numbers K 4± (c,ε,ρ,α ± ,n,k ± =|k ± η (c) k± ,n(ε,ρ,α ± ) and M 4± (c,ε,ρ,α ± ,n,k ± )= |a ± |η (c) k± ,n(ε,ρ,α ± ) are constructed in which η k± , (c) n(ε,ρ,α ± ) are the n th positive purely imaginary zeros of the F 4 functions in x, n=1,2,3, sgn k= sgn α and the parameters acquire the values pointed out. The class of L 4 numbers, involving the subclasses L a 4± = L a 4± (c,ε a ,ρ,α ± ,n), L̅ 4± = L̅ 4± (c,̅ε,ρ,α ± ,n), L̃ 4± = L̃ 4± (c,̅ε,ρ,α ± ,n), L̃ 4± = L̃ 4± (c,̅ε,ρ,α ± ,n) and L̃ 4lim± = L̃ 4lim± (a,c;x,ε̃,ρ,α) is advanced. Each of its elements equals the common limits of the sequences mentioned, put together of the zeros of the respective F 4 function, attained, provided k + → +∞ or K - → -∞ (the subscripts "+", "-" correspond to the relevant sign of k). Some values of the quantities L 4 are computed and presented in a tabular form. The significance of the numbers considered for the theory of waveguides is debated.]]></abstract><pub>IEEE</pub></addata></record> |
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| subjects | Antenna arrays Bars Equations Mathematical model Phase shifters Physics Waveguide theory |
| title | On the L4 numbers and their application |
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