Partial-inductance retarded partial coefficients: Their exact computation based on the Cagniard-DeHoop technique

Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91 A ailable online 20 Janua y 2023 0955-7997/© 2023 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). Con en s lis s a ailable a ScienceDi ec Enginee ing Analysis wi h Bounda y Elemen s jou nal homepage: www.else ie .com/loca e/enganabound Pa ial-induc ance e a ded pa ial coe icien s: Thei exac compu a ion based on he Cagnia d–DeHoop echnique Ma in S ump a, Fab izio Lo e ob,∗, Giuseppe Pe anicec, Giulio An oninib aLe ch Labo a o y o EM Resea ch, Depa men Radio Elec onics, FEEC, B no Uni e si y o Technology, B no, Czech Republic bDepa men o Indus ial and In o ma ion Enginee ing and Economics, Uni e si à degli S udi dell’Aquila, L’Aquila, I aly cDepa men o Enginee ing and In o ma ion Sciences and Ma hema ics, Uni e si à degli S udi dell’Aquila, L’Aquila, I aly ARTICLE INFO Keywo ds: Pa ial Elemen Equi alen Ci cui me hod Cagnia d–DeHoop echnique Compu a ional elec omagne ics Pa ial induc ances Time-domain modeling ABSTRACT The Pa ial Elemen Equi alen Ci cui (PEEC) me hod is a well ecognized in eg al-equa ion (IE) echnique o sol e Maxwell’s equa ions. Simila ly o he me hod o momen s (MoM), he elec omagne ic (EM) in e ac ions be ween cu en s and be ween cha ges a e desc ibed in e ms o in eg als. In con as o he s anda d MoM, he PEEC me hod keeps he elec ic and magne ic coupling phenomena sepa a e, which leads o di e en in e ac ion in eg als o be compu ed. These in eg als admi simpli ied solu ions o he case o he s a ic ee-space G een’s unc ion and o hogonal geome ies bu hei applicabili y is limi ed o elec ically small p oblems only. When he ull-wa e ee-space G een’s unc ion is conside ed, he in eg als a e ypically compu ed in he equency domain (FD) by eso ing o Gaussian quad a u e schemes. The accu acy and e iciency o such schemes is a delica e issue. The e o e, ecen wo ks ha e in es iga ed he possibili y o applying he Cagnia d–DeHoop (CdH) echnique o calcula e he in e ac ion in eg als o ze o- hickness elemen a y domains. In his pape , we close he loop and shall apply he CdH echnique o calcula e he pa ial-induc ance be ween wo elemen a y b icks as p esc ibed by he PEEC echnique exac ly in he ime domain (TD). The analy ical app oach is demons a ed on he in e ac ion be ween wo b icks as i occu s in he modeling o he magne ic ield coupling be ween olume ic cu en s. The accu acy o he p oposed app oach is (success ully) es ed o wo ep esen a i e cases. 1. In oduc ion The no ion o pa ial induc ance, as in oduced by D . A. E. Ruehli in 1972 [1], is a undamen al concep on which he PEEC me hod lays i s ounda ions. The PEEC me hod is an IE me hod capable o analyzing an EM sca e ing p oblem by means o an equi alen ci cui ep esen a ion [2]. The PEEC me hod can be o mula ed wi h he aid o s anda d EM con as -sou ce in eg al ep esen a ions [3, Sec. 28.9], in which he G een’s unc ions apply o he ( ypically homogeneous, iso opic and loss- ee) medium o he embedding. Hence, he equi - alen ci cui can be di ec ly associa ed wi h a disc e ized sca e e , whe e elemen a y olumes and su aces a e assumed o adia e in he backg ound medium (e.g. he ee-space). The PEEC me hod has been p ima ily employed in he solu ion o FD p oblems and, in his con ex , in e ac ions in eg als ha e been compu ed ei he making quasi-s a ic assump ions (e.g., neglec ing he e a da ion e ms in he in eg als) [4] o by eso ing o quad a u e schemes in he equency domain (FD) [5] o by means o he Taylo expansion o he G een’s unc ion [6]. Simi- la lines o easoning a e commonly ollowed o ex ac he singula i y ∗Co esponding au ho . E-mail add ess: [email p o ec ed] (F. Lo e o). o G een’s unc ion in s anda d in eg al-equa ion o mula ions (e.g., [7– 9]). Owing o he e e inc easing in e es in accu a e TD simula ions in he ield o Elec omagne ic Compa ibili y (EMC), howe e , ull- wa e TD o mula ions o he pa ial induc ance a e becoming mo e and mo e impo an . Acco dingly, his issue has been ecen ly add essed analy ically o ela i ely simple 2-D ze o- hickness pa ches [10,11]. In his a icle, we shall apply he CdH echnique [12] o calcu- la e a pa ial-induc ance e a ded PEEC coe icien exac ly in he TD. The p esen ed analy ical esul s apply o he in e ac ion be ween wo b icks (= igh pa allelepipeds). The CdH echnique is a join - ans o m me hod ha has been o iginally de eloped o analy ically analyze he seismic-wa e p opaga ion in ho izon ally laye ed media (e.g. [13, 14]). Mo e ecen ly, i has been demons a ed ha his sophis ica ed in e sion me hodology can also be use ul o cons uc ing pu ely nu- me ical solu ions. Indeed, he CdH echnique is a key ing edien in he Cagnia d–DeHoop me hod o momen s, a no el TD-IE echnique o he TD analysis o EM adia ion and sca e ing p oblems [15,16]. Mo e- o e , ou ini ial s udies (see [10,11]) analyzing ze o- hickness PEEC h ps://doi.o g/10.1016/j.enganabound.2023.01.008 Recei ed 9 Augus 2022; Recei ed in e ised o m 8 Janua y 2023; Accep ed 8 Janua y 2023 Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91 87 M. S ump e al. Fig. 1. Two in e ac ing b ick elemen s. coe icien s ha e demons a ed ha he CdH in e sion is a p omising s a egy o achie ing hei analy ical exp essions in he TD. Acco d- ingly, his pape epo s on he ecen esul s o ou e o s o de elop a quasi-s a ic-app oxima ion- ee PEEC sol e ha is equipped wi h exac , CdH-based TD olume ic pa ial-induc ance. I is an icipa ed ha he p esen ed app oach can be employed along wi h he p e- co ec ed as Fou ie ans o m (FFT) [17] and he FFT-based app oach o accele a e he ma ix– ec o p oduc s [18–20]. The pape is o ganized as ollows: in Sec ion 2 he p oblem o mu- la ion is p esen ed. Consequen ly, he TD analy ical solu ion is gi en in Sec ion 3, which is supplemen ed wi h Appendix. Finally, in Sec ion 4, h ee nume ical applica ions a e p esen ed and success ully alida ed. 2. P oblem o mula ion A PEEC model is ep esen ed h ough a se o pa ial elemen s, he alue o which is ound upon e alua ing spa ial in eg als o e he su aces/ olumes o in e ac ing disc e iza ion elemen s. In his wo k we shall analyze he in e ac ion o wo b icks (= igh pa allelepipeds) (see Fig. 1). In pa icula , wi h e e ence o [21, Eq. (4)], we shall s udy a e a ded pa ial po en ial coe icien exp essed h ough a double in eg al  𝐿𝑚𝑛(𝑠) = 𝜇0 𝑚𝑛∫𝒓∈𝑚 d𝑉∫𝒓′∈𝑛 𝑔(𝒓−𝒓′, 𝑠)d𝑉′,(1) whe e 𝑚= {−𝛥𝑚 𝑥∕2 < 𝑥 −𝑥𝑚< 𝛥𝑚 𝑥∕2,−𝛥𝑚 𝑦∕2 < 𝑦 −𝑦𝑚< 𝛥𝑚 𝑦∕2,−𝛥𝑚 𝑧∕2 < 𝑧−𝑧𝑚< 𝛥𝑚 𝑧∕2} and 𝑛= {−𝛥𝑛 𝑥∕2 < 𝑥 −𝑥𝑛< 𝛥𝑛 𝑥∕2,−𝛥𝑛 𝑦∕2 < 𝑦 −𝑦𝑛< 𝛥𝑛 𝑦∕2,−𝛥𝑛 𝑧∕2 < 𝑧 −𝑧𝑛< 𝛥𝑛 𝑧∕2}, whe e 𝛥𝑚,𝑛 𝑥>0,𝛥𝑚,𝑛 𝑦>0and 𝛥𝑚,𝑛 𝑧>0 deno e he spa ial disc e iza ion s eps in he 𝑥-, 𝑦- and 𝑧-di ec ion, espec i ely. Fu he mo e, 𝑠is he Laplace- ans o m pa ame e wi h Re(𝑠)>0, and 𝑚,𝑛 a e c oss sec ions o he olumes 𝑚,𝑛, espec- i ely, ha a e pe pendicula o he co esponding elec ic-cu en lows. Nex , 𝑔(𝒓, 𝑠) = exp(−𝑠|𝒓|∕𝑐) 4𝜋|𝒓|(2) is he ee-space G een’s unc ion o he 3-D scala modi ied Helmhol z equa ion and 𝑐= (𝜀𝜇)−1∕2 >0deno es he pe inen ( eal- alued and posi i e) EM wa e speed. 3. P oblem solu ion The e a ded pa ial po en ial coe icien as exp essed h ough Eq. (1) will be nex ans o med o he TD analy ically wi h he aid o he CdH echnique. Pu suing his app oach and assuming ha Fig. 2. Two in e ac ing cube elemen s. |𝑧𝑚−𝑧𝑛|>(𝛥𝑚 𝑧+𝛥𝑛 𝑧)∕2, one may exp ess he TD o iginal o Eq. (1), u he deno ed by 𝐿𝑚𝑛(𝑡)(in hen y/second = ohm), as ollows: 𝐿𝑚𝑛(𝑡)=(𝜇0∕𝑚𝑛)[𝐽(|𝑧𝑚−𝑧𝑛|+𝛥𝑚𝑛+ 𝑧, 𝑡) −𝐽(|𝑧𝑚−𝑧𝑛|+𝛥𝑚𝑛− 𝑧, 𝑡) − 𝐽(|𝑧𝑚−𝑧𝑛|−𝛥𝑚𝑛− 𝑧, 𝑡) +𝐽(|𝑧𝑚−𝑧𝑛|−𝛥𝑚𝑛+ 𝑧, 𝑡)],(3) whe e 𝐽(𝑧, 𝑡) = 𝐼(𝑥𝑚−𝑥𝑛+𝛥𝑚𝑛+ 𝑥, 𝑧, 𝑡) −𝐼(𝑥𝑚−𝑥𝑛+𝛥𝑚𝑛− 𝑥, 𝑧, 𝑡) − 𝐼(𝑥𝑚−𝑥𝑛−𝛥𝑚𝑛− 𝑥, 𝑧, 𝑡) +𝐼(𝑥𝑚−𝑥𝑛−𝛥𝑚𝑛+ 𝑥, 𝑧, 𝑡),(4) and 𝐼(𝑥, 𝑧, 𝑡) = 𝐾(𝑥, 𝑦𝑚−𝑦𝑛+𝛥𝑚𝑛+ 𝑦, 𝑧, 𝑡) −𝐾(𝑥, 𝑦𝑚−𝑦𝑛+𝛥𝑚𝑛− 𝑦, 𝑧, 𝑡) −𝐾(𝑥, 𝑦𝑚−𝑦𝑛−𝛥𝑚𝑛− 𝑦, 𝑧, 𝑡) +𝐾(𝑥, 𝑦𝑚−𝑦𝑛−𝛥𝑚𝑛+ 𝑦, 𝑧, 𝑡),(5) whe e we used 𝛥𝑚𝑛± 𝑥,𝑦,𝑧 = (𝛥𝑚 𝑥,𝑦,𝑧 ±𝛥𝑛 𝑥,𝑦,𝑧)∕2,(6) espec i ely. He e, 𝐾(𝑥, 𝑦, 𝑧, 𝑡) ep esen s he TD o iginal o he gene ic slowness in eg al, he de ini ion and in e sion o which is p esen ed in Appendix. Finally, we emphasize ha Eq. (3) applies o he con- igu a ion whe e |𝑧𝑚−𝑧𝑛|> 𝛥𝑚𝑛+ 𝑧. The case |𝑧𝑚−𝑧𝑛|< 𝛥𝑚𝑛+ 𝑧mus be analyzed sepa a ely. 4. Nume ical examples In his sec ion, nume ical examples ela ed o h ee di e en ge- ome ies a e p esen ed. Fo alida ion pu poses, he esul s ob ained h ough he p oposed me hod a e compa ed wi h hose ob ained h ough he nume ical-in e sion o he Laplace ans o m (NILT) ap- p oach [22–24]. 4.1. Two in e ac ing cubes As a pa icula applica ion, he esul ing TD exp ession (3) has been implemen ed in MATLAB®and applied o he case o wo in e ac ing cubes (see Fig. 2). The e alua ions a e pe o med in he ini e ime window {0 ≤𝑐𝑡∕𝑅𝑚𝑛 ≤2}, whe e 𝑅𝑚𝑛 = [(𝑥𝑚−𝑥𝑛)2+ (𝑦𝑚−𝑦𝑛)2+ (𝑧𝑚− 𝑧𝑛)2]1∕2 ep esen s he cen e - o-cen e dis ance be ween wo (iden ical) cubes loca ed a •(𝑥𝑚, 𝑦𝑚, 𝑧𝑚) = (0,0,0), Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91 88 M. S ump e al. Fig. 3. The TD coe icien o he wo iden ical cubes: compa ison be ween he p oposed echnique and he dNILT2 - He mi e echnique.. Fig. 4. Spec um compa ison o he in e ac ion be ween wo iden ical cubes. •(𝑥𝑛, 𝑦𝑛, 𝑧𝑛) = (2𝛥𝑥,2𝛥𝑥,2𝛥𝑥). In he p esen example we ake 𝛥𝑥=𝛥𝑚,𝑛 𝑥=𝛥𝑚,𝑛 𝑦=𝛥𝑚,𝑛 𝑧= 1.0 mm. The esul ing pulse shape o he TD coe icien is shown in 3. He e, o he sake o alida ion, he esul s ob ained h ough he p oposed CdH- based and he ( e e en ial) dNILT2 - He mi e echnique a e p esen ed. Fo a de ailed desc ip ion o he e e en ial me hodology we e e he eade o [24]. The ci cle poin s a e he ini ial samples needed o he in e polan building. Finally, he FD coun e pa o he wo TD esponses is ske ched in Fig. 4. As can be seen, he compu ed esul s show good co espondence wi h dissimila i ies occu ing om 400 GHz up. These disc epancies a e, howe e , i ually negligible in echnical applica ions. 4.2. In e ac ions o a sys em o cubes As a u he example we conside he geome y depic ed in Fig. 5, whe e he mu ual pa ial induc ance be ween he cube wi h he cen e loca ed a he axes o igin and he o he s a e conside ed. All he cubes ha e sides 𝛥𝑥=𝛥𝑦=𝛥𝑧= 1.0 mm. The cubes sys em is composed by 27 elemen s, h ee o each dimension. The cen e coo dina es span a ange [2𝛥𝑥− 8𝛥𝑥]. In Fig. 6 a e ske ched all he TD coe icien s ela ed o he sys em, each one compu ed h ough he p oposed echnique and compa ed o he dNILT2 - He mi e echnique [24]. The la e Fig. 5. Cubes sys em geome y. Fig. 6. TD coe icien s o he cubes sys em geome y. echnique is based on he cons uc ion o an accu a e in e pola o , s a ing om he knowledge o he alues o he unc ion and i s i s highe o de de i a i es a he s a ing poin s, enci cled in ed in he igu e (gene ally up o he i h o de is su icien ). 4.3. Induced ol age on a cube by he cu en s lowing in ou b icks In Fig. 7 a e shown ou iden ical pa allelepipeds: 1,2,3,4, wi h sides: 𝛥𝑥= 1.5 mm, 𝛥𝑦= 0.5 mm, 𝛥𝑧= 0.25 mm, abo e a cube 0, wi h sides: 𝛥0 𝑥=𝛥0 𝑦=𝛥0 𝑧= 1.0 mm, and cen e loca ed a he axes o igin. The pa allelepipeds a e conside ed a he same heigh and a e colloca ed unsymme ically in he 𝑥–𝑦plane, wi h espec o he cube. In pa icula , he coo dina es o he pa allelepipeds a e: •(𝑥1, 𝑦1, 𝑧1) = (−2.5 mm,0.5 mm,3𝛥0 𝑥), •(𝑥2, 𝑦2, 𝑧2) = (−2.5 mm,5 mm,3𝛥0 𝑥), •(𝑥3, 𝑦3, 𝑧3) = (11 mm,0.5 mm,3𝛥0 𝑥), •(𝑥4, 𝑦4, 𝑧4) = (11 mm,5 mm,3𝛥0 𝑥). The o e all induced ol age on he cube 0by he sys em o pa al- lelepipeds can be compu ed h ough a combina ion o ou con olu ion in eg als as: 𝑣𝐿0(𝑡) = 4 ∑ 𝑛=1 ∫𝑡 0 𝐿𝑝0,𝑛 (𝑡−𝜏)d𝑖𝑛(𝜏) d𝜏d𝜏(7) Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91 89 M. S ump e al. Fig. 7. Geome y o he compu a ion o he induced ol age on a cube by he cu en s lowing in he b icks sys em. Fig. 8. Induced ol age on he cube by he cu en s lowing in he sys em o b icks. whe e 𝑖𝑛(𝑡)is he imp essed cu en on each pa allelepiped, 𝑛= 1, ⋯,4, lowing in he 𝑥di ec ion. The imp essed cu en 𝑖𝑛(𝑡), o each pa allelepiped, is assumed o exhibi a windowed-powe (WP) wa e o m [25]: 𝑖𝑛(𝜏, 𝑡) = 𝑡′𝜏(2 − 𝑡′)𝜏H(𝑡′)H(2 − 𝑡′)(8) whe e H(𝑡)is he Hea iside uni -s ep unc ion (H(𝑡)=0i 𝑡 < 0, H(0) = 1∕2,H(𝑡) = 1 i 𝑡 > 0), 𝑡′=𝑡∕𝑡 ,𝑡 being he pulse ise ime. We choose 𝜏= 2 and 𝑡 = 4.6ps. The induced ol age on he cube 0is depic ed in Fig. 8, whe e i is obse ed an excellen ag eemen be ween he CdH me hod and he NILT-based me hod. 5. Conclusions The PEEC me hod equi es ha in e ac ion in eg als desc ibing he magne ic ield coupling be ween elemen a y olume ic egions be compu ed, namely pa ial induc ances. In he equency domain, his is usually done by eso ing o quad a u e schemes. In he TD, he use o o e -simpli ying assump ions on he p opaga ion delay leads o app oxima e esul s wi h a nega i e impac on physical p ope ies such as he causali y and s abili y o he model. In his wo k, quasi- closed- o m o TD e a ded pa ial induc ances ha e been de i ed using he Cagnia d–DeHoop (CdH) echnique. A pe inen in eg a ion pa h de o ma ion in he complex slowness plane allows o ob ain semi- analy ical o ms o he ansien in e ac ion in eg als o a pai o o hogonal b icks as hey occu in he PEEC me hod using Manha an- ype meshes o oxelliza ion echniques. The p oposed app oach has been es ed o ep esen a i e es cases by compa ison wi h o he nume ical me hods, always exhibi ing a e y good ag eemen . Fig. 9. Complex slowness planes. (a) 𝜎-plane wi h he CdH-pa h o 𝑦 < 0; (b) 𝜅-plane wi h he CdH-pa hs o 𝑥 < 0. Decla a ion o compe ing in e es The au ho s decla e ha hey ha e no con lic o in e es . Da a a ailabili y The p esen ed da a a e a ailable upon eques om he au ho s. Acknowledgmen s The esea ch o Ma in S ump was suppo ed by he Czech Science Founda ion unde G an No. 20-01090S. Appendix. The gene ic in eg al The in eg al ep esen a ion o be ans o med o TD has he ollow- ing o m  𝐾(𝑥, 𝑦, 𝑧, 𝑠) = (𝑠 2i𝜋)2∫𝜅∈K0 exp(𝑠𝜅𝑥) 𝑠2𝜅2d𝜅 ×∫𝜎∈S0 exp{−𝑠[−𝜎𝑦 +𝛤(𝜅, 𝜎)𝑧]} 𝑠2𝜎2 d𝜎 2𝑠3𝛤3(𝜅, 𝜎)(9) o 𝑥∈R,𝑦∈R,{𝑧∈R;𝑧≥0} and {𝑠∈R;𝑠 > 0}, whe e K0and S0a e he in eg a ion pa hs ex ending along Re(𝜅)=0and Re(𝜎)=0, espec i ely, ha a e inden ed o he igh wi h semi-ci cula a cs wi h cen e s a he o igins and anishingly small adii (see Fig. 9). Finally, 𝛤(𝜅, 𝜎), being he slowness pa ame e along he 𝑧-di ec ion, is de ined as 𝛤(𝜅, 𝜎) = (1∕𝑐2−𝜅2−𝜎2)1∕2 wi h Re(𝛤)≥0.(10) The gene ic in eg al will nex be ans o med o he TD wi h he aid o he CdH echnique (see [12] and [16, Ch. 2]). To ha end, he in eg a ion con ou in he complex 𝜎-plane, S0, is by i ue o Jo dan’s lemma and Cauchy’s heo em [3, p. 1054] de o med in o a CdH pa h, say ∪∗(he e ∗deno es he complex conjuga e), along Enginee ing Analysis wi h Bounda y Elemen s 149 (2023) 86–91 90 M. S ump e al. which −𝜎𝑦 +𝛤(𝜅, 𝜎)𝑧=𝑢𝑑𝛺(𝜅) o {1 ≤𝑢 < ∞} wi h 𝑑2=𝑦2+𝑧2 and 𝛺(𝜅) = (1∕𝑐2−𝜅2)1∕2 is sa is ied (see Fig. 9a). Upon combining he con ibu ions om and ∗, he inne in eg al wi h espec o 𝜎can be cas in o he in eg al wi h espec o he ( eal- alued and posi i e) pa ame e s 𝑢. In addi ion, he con ibu ion om he (double) pole singula i y a 𝜎= 0 mus be o 𝑦 > 0accoun ed o . The hus exp essed inne in eg al is subsequen ly subs i u ed back in Eq. (9), which yields  𝐾(𝑥, 𝑦, 𝑧, 𝑠) =  𝑀(𝑥, 𝑦, 𝑧, 𝑠) +  𝑁(𝑥, 𝑦, 𝑧, 𝑠),(11) whe e  𝑀=1 2𝜋i 𝑑4 2𝜋𝑠3∫∞ 𝑢=1 𝑦2𝑧2−𝑢2(𝑢2− 1)(𝑦4− 6𝑦2𝑧2+𝑧4) (𝑢2𝑑2−𝑦2)2(𝑢2𝑑2−𝑧2)2 ×d𝑢 (𝑢2− 1)1∕2 ∫𝜅∈K0 exp{−𝑠[−𝜅𝑥 +𝛺(𝜅)𝑢𝑑]} d𝜅 𝑠2𝜅2𝛺4(𝜅)(12) and  𝑁=1 2𝜋i 𝑦H(𝑦) 2𝑠2∫𝜅∈K0 exp{−𝑠[−𝜅𝑥 +𝛺(𝜅)𝑧]} d𝜅 𝑠2𝜅2𝛺3(𝜅),(13) whe e H(𝑦)has again he meaning o he Hea iside uni -s ep unc ion, i.e. H(𝑦)=0i 𝑦 < 0,H(0) = 1∕2,H(𝑦)=1i 𝑦 > 0. Fi s , we shall desc ibe he ans o ma ion o  𝑀as gi en by Eq. (12). Fo his pu pose, he in eg a ion con ou in he complex 𝜅-plane, K0, is de o med in o a CdH pa h, say ∪∗, along which −𝜅𝑥+𝛺(𝜅)𝑢𝑑 =𝜏 o {𝑅(𝑢)∕𝑐≤𝜏 < ∞} wi h 𝑅(𝑢)=(𝑥2+𝑢2𝑑2)1∕2 >0is sa is ied (see Fig. 9b). In he esul ing exp ession, we combine he con ibu ions om and ∗and change he o de o he in eg a ions acco ding o, symbolically ∫∞ 𝑢=1 d𝑢∫∞ 𝜏=𝑅(𝑢)∕𝑐 d𝜏→∫∞ 𝜏=𝑅(1)∕𝑐 d𝜏∫𝑈(𝑐𝜏) 𝑢=1 d𝑢(14) whe e 𝑈(𝑐𝜏) = (𝑐2𝜏2∕𝑑2−𝑥2∕𝑑2)1∕2. Upon ca ying ou he in eg a ion wi h espec o 𝑢, Eq. (12) can be cas in o he ollowing o m  𝑀=𝑐6 2𝜋2𝑠5∫∞ 𝜏=𝑅(1)∕𝑐 exp(−𝑠𝜏)(𝑥, 𝑦, 𝑧, 𝑐𝜏)d𝜏 + 𝑃(𝑥, 𝑦, 𝑧, 𝑠)(15) whe e  𝑃a ises om he (double) pole singula i y a 𝜅= 0. Bo h e ms on he igh -hand side o Eq. (15) ha e he o m ha allows hei s aigh o wa d ans o m o he o iginal domain. In his p ocedu e, Le ch’s uniqueness heo em applying o he eal- alued and posi i e Laplace- ans o m pa ame e is an essen ial esul ha we ely on [26, Appendix]. The ans o ma ion o  𝑁(see Eq. (13)) ollows simila lines o easoning. Indeed, he o iginal in eg a ion con ou , K0, is i s eplaced wi h a new CdH pa h along which −𝜅𝑥+𝛺(𝜅)𝑧=𝜏is me o all {𝜌∕𝑐≤ 𝜏 < ∞}, whe e 𝜌2=𝑥2+𝑧2. Combining again he con ibu ions om he hype bolic a cs in he lowe and uppe hal es o he complex 𝜅-plane, we end up wi h an in eg al wi h espec o 𝜏 ha can be exp essed as  𝑃(𝑦, 𝑥, 𝑧, 𝑠)(c . Eq. (15)). Rep esen ing u he he con ibu ion om he (double) pole singula i y a 𝜅= 0 by  𝑄(𝑥, 𝑦, 𝑧, 𝑠), we a i e a  𝑁= 𝑃(𝑦, 𝑥, 𝑧, 𝑠) +  𝑄(𝑥, 𝑦, 𝑧, 𝑠).(16) Upon subs i u ing Eqs. (15) wi h (16) in (11) and ans o m he esul o he TD, we inally ge 𝐾(𝑥, 𝑦, 𝑧, 𝑡) = 𝑐 48𝜋2∫𝑐𝑡 𝑣=𝑅 (𝑐𝑡 −𝑣)4(𝑥, 𝑦, 𝑧, 𝑣)d𝑣 +𝑃(𝑥, 𝑦, 𝑧, 𝑡) + 𝑃(𝑦, 𝑥, 𝑧, 𝑡) + 𝑄(𝑥, 𝑦, 𝑧, 𝑡).(17) The unc ion behind he in eg al sign is gi en by (𝑥, 𝑦, 𝑧, 𝑣) = ∫𝜋∕2 𝜓=0 𝑓(𝑥, 𝑦, 𝑧, 𝑣, 𝜓) ×𝑝2−𝑞2(𝑈2− 1) sin2(𝜓)[cos2(𝜓) + 𝑈2sin2(𝜓)] {(𝑧2∕𝑑2) cos2(𝜓) + [(𝑣2−𝑟2)∕𝑑2] sin2(𝜓)}2 ×d𝜓 {(𝑦2∕𝑑2) cos2(𝜓) + [(𝑣2−𝜌2)∕𝑑2] sin2(𝜓)}2,(18) wi h 𝑟2=𝑥2+𝑦2,𝑝2=𝑦2𝑧2∕𝑑4,𝑞2=𝑦4∕𝑑4− 6𝑝2+𝑧4∕𝑑4,𝑈2= 𝑣2∕𝑑2−𝑥2∕𝑑2and 𝑓=𝑣∕𝑑2 𝑈3[(𝑥2∕𝑑2) sin2(𝜓)+(𝑣2∕𝑑2− 1) cos2(𝜓)]2 ×{(3𝑥2 𝑑2+𝑣2 𝑑2)[cos2(𝜓) + 𝑈2sin2(𝜓)]3 +(4𝑥4 𝑑4−11𝑥2𝑣2 𝑑4−𝑣4 𝑑4)[cos2(𝜓) + 𝑈2sin2(𝜓)]2 −(𝑥6 𝑑6+5𝑥4𝑣2 𝑑6− 10 𝑥2𝑣4 𝑑6)[cos2(𝜓) + 𝑈2sin2(𝜓)] − 2 𝑥8 𝑑8+ 7 𝑥6𝑣2 𝑑8− 5 𝑥4𝑣4 𝑑8}.(19) The emaining e ms in he inal TD esul (17) can be exp essed as ollows 𝑃(𝑥, 𝑦, 𝑧, 𝑡) = 𝑐𝑥𝑑3H(𝑥) 12𝜋{6|𝑦|𝑧𝑐𝑡 𝑑3 ×{ an−1 [|𝑦|(𝑐2𝑡2−𝑑2)1∕2 𝑧𝑐𝑡 ] + an−1 [𝑧(𝑐2𝑡2−𝑑2)1∕2 |𝑦|𝑐𝑡 ]} − 3 𝑧 𝑑 𝑐2𝑡2+𝑦2 𝑑2 an−1 [(𝑐2𝑡2−𝑑2)1∕2 𝑧] − 3 |𝑦| 𝑑 𝑐2𝑡2+𝑧2 𝑑2 an−1 [(𝑐2𝑡2−𝑑2)1∕2 |𝑦|] +(2𝑐2𝑡2 𝑑2+ 1)(𝑐2𝑡2 𝑑2− 1)1∕2},(20) and, inally, 𝑄(𝑥, 𝑦, 𝑧, 𝑡) = 𝑐𝑥𝑦H(𝑥)H(𝑦) 4(𝑐𝑡 −𝑧)2H(𝑐𝑡 −𝑧).(21) Re e ences [1] Ruehli AE. 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