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Journal articles on the topic 'LEGENDERS FUNCTIONS'

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Published: 11 September 2023

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1

van der Toorn, Ramses. "The Singularity of Legendre Functions of the First Kind as a Consequence of the Symmetry of Legendre’s Equation." Symmetry 14, no. 4 (April 4, 2022): 741. http://dx.doi.org/10.3390/sym14040741.

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Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of the first and second kind. In physics, however, commonly the only acceptable members of this set are Legendre polynomials. The quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone and in-depth argument for rejecting all solutions of Legendre’s equation in physics apart from the polynomial ones. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation are quintessential to the argument. We demonstrate that the evenness or oddness of Legendre polynomials is a consequence of the same premises.

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2

Darlam Bagus, Andreow Kony. "STRUKTUR NARASI, FUNGSI DAN NILAI BUDAYA DALAM KUMPULAN LEGENDA DI KALIMANTAN TENGAH." JURNAL BAHASA, SASTRA DAN PEMBELAJARANNYA 9, no. 2 (December 7, 2019): 112. http://dx.doi.org/10.20527/jbsp.v9i2.7475.

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Narrative Structure, Function, and Cultural Values in Legends of CentralKalimantan. Research entitled is motivated by the interest of researcher topreserve the legend of the area that is now almost extinct and forgotten by thecommunity. This study aims to describe (1) the structure of narrative in legends ofcentral kalimantan, (2) the function of legend in legendsof central kalimantan, (3)cultural values in legends of central kalimantan. This research used a descriptivemethod. the data source can be from fifteen legends in central kalimantan creationby Prof. Drs. Kumpiady Widen, MA, Ph. D then collected through severaltechniques, classified, analyzed, and concluded. Based on the results of dataanalysis, it can be concluded that of the fifteen legends that have been analyzed,there are ten local legends and five individual legends. Based on analysis of thenarrative structure, the five narrative stages are found from the fifteen legends.Based on analysis of legend’s function, the six functions of legend used as theory inthis study, only found four functions of legend, the function of legend to help theeducation of young people, increase feelings of solidarity of group, socialsanctions for people who behave well or punish, and last as a means of socialcriticism. based on analysis of the cultural values contained in these fifteen legendsfound five cultural values, namely: the cultural values of human life, the culturalvalues of human works, the cultural values of human perceptions of time, thecultural values of the human view of nature and the latter are cultural valueshuman relationships with others. Key words: naration structure, legendary function, culture values Abstrak Struktur Narasi, Fungsi, dan Nilai Budaya dalam Kumpulan Legenda diKalimantan Tengah. Penelitian ini dilatarbelakangi oleh ketertarikan penelitiuntuk melestarikan legenda daerah yang sekarang sudah hampir punah dandilupakan oleh masyarakat. Penelitian ini bertujuan mendeskripsikan (1) Strukturnarasi dalam kumpulan legenda di Kalimantan Tengah (2) Fungsi legenda dalamkumpulan legenda di Kalimantan Tengah (3) Nilai Budaya dalam kumpulanlegenda di Kalimantan Tengah. Penelitian ini menggunakan metode deskriptif. Sumber data didapat dari lima belas legenda yang ada di Kalimantan Tengahkarya Prof. Drs. Kumpiady Widen, MA, Ph. D. Berdasarkan analisis mengenaistruktur narasi, kelima tahap narasi ditemukan dari lima belas legenda tersebut.Berdasarkan analisis mengenai fungsi legenda, dari enam fungsi legenda yang dijadikan teori dalam penelitian ini hanya ditemukan empat fungsi legenda.Berdasarkan analisis mengenai nilai budaya yang terkandung dalam kelima belaslegenda tersebut ditemukan lima nilai budaya, yaitu nilai budaya tentang hidupmanusia, nilai budaya tentang karya manusia, nilai budaya persepsi manusiatentang waktu, nilai budaya pandangan manusia terhadap alam dan yang terakhirnilai budaya hubungan manusia dengan sesama. Kata-kata kunci: struktur narasi, fungsi legenda, nilai budaya

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3

Khan, Nabiullah U., and Talha Usman. "Certain generating functions of Hermite - Bernoulli - Legendre polynomials." Ufimskii Matematicheskii Zhurnal 10, no. 2 (2018): 118–26. http://dx.doi.org/10.13108/2018-10-2-118.

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4

Chan, Fuk‐Lok, Kim‐Ming Lau, and Robert J. Finkelstein. "Grassmannian Legendre functions." Journal of Mathematical Physics 33, no. 8 (August 1992): 2688–96. http://dx.doi.org/10.1063/1.529590.

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5

Jones, D. L., and C. P. Burke. "Zonal harmonic series expansions of Legendre functions and associated Legendre functions." Journal of Physics A: Mathematical and General 23, no. 14 (July 21, 1990): 3159–68. http://dx.doi.org/10.1088/0305-4470/23/14/015.

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6

Mereu, Myriam. "Cogas, janas e le altre: le creature mitiche e fantastiche nella letteratura e nel cinema sardi." Italianistica Debreceniensis 24 (December 1, 2018): 56–76. http://dx.doi.org/10.34102/italdeb/2018/4661.

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Sardinian contemporary literature and films have recently recovered an extensive heritage of folk myths and legends taken from the oral tradition. Legendary figures, such as accabadoras (female figure who was enabled with the task of easing the sufferings of the dying people), and fantasy creatures, such as cogas, surbiles (‘vampire witches’), janas (‘fairies, pixies’), and panas (‘the ghosts of women who died in childbirth’) are being revived by writers and film directors with the purpose to bring their memory back to life and share it with a wide audience of readers and spectators. The analysis of imaginary and legendary creatures in Sardinian contemporary literature cannot overlook orality and its central role in shaping popular imagination over the centuries. Writing has replaced orality, whilst mass media and digital media are getting the upper hand over storytelling as a practice of community and family aggregation, meant to mark the long working hours and scare the children, amongst the most common functions of Sardinian oral storytelling. The literary corpus includes fairy tales, novels, tales and legends dealing with the Sardinian oral tradition, whilst on the cinematic side I will examine short films, feature films and documentaries made in Sardinia over the last fifteen years.

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7

Love. "EXPANSION VIA LEGENDRE FUNCTIONS." Real Analysis Exchange 19, no. 1 (1993): 50. http://dx.doi.org/10.2307/44153808.

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8

Babich, V. M., D. B. Dement’ev, and B. A. Samokish. "Computation of Legendre functions." Journal of Mathematical Sciences 148, no. 5 (February 2008): 639–49. http://dx.doi.org/10.1007/s10958-008-0014-3.

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9

Jung, Soon-Mo. "Approximation of analytic functions by Legendre functions." Nonlinear Analysis: Theory, Methods & Applications 71, no. 12 (December 2009): e103-e108. http://dx.doi.org/10.1016/j.na.2008.10.007.

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10

Lohöfer, G. "Inequalities for Legendre functions and Gegenbauer functions." Journal of Approximation Theory 64, no. 2 (February 1991): 226–34. http://dx.doi.org/10.1016/0021-9045(91)90077-n.

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11

JONES, D. S. "SOME PROPERTIES OF LEGENDRE FUNCTIONS." Analysis and Applications 02, no. 02 (April 2004): 129–43. http://dx.doi.org/10.1142/s0219530504000035.

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Some integral representations involving Legendre functions are given. Some are related to Bessel functions and their products. Others are concerned with zeros of Legendre functions and may provide a suitable basis for computational purposes.

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12

Lal, S., and V. K. Sharma. "On the estimation of functions belonging to Lipschitz class by block pulse functions and hybrid Legendre polynomials." Carpathian Mathematical Publications 12, no. 1 (June 12, 2020): 111–28. http://dx.doi.org/10.15330/cmp.12.1.111-128.

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In this paper, block pulse functions and hybrid Legendre polynomials are introduced. The estimators of a function $f$ having first and second derivative belonging to $Lip_\alpha[a,b]$ class, $0 < \alpha \leq 1$, and $a$, $b$ are finite real numbers, by block pulse functions and hybrid Legendre polynomials have been calculated. These calculated estimators are new, sharp and best possible in wavelet analysis. An example has been given to explain the validity of approximation of functions by using the hybrid Legendre polynomials approximation method. A real-world problem of radioactive decay is solved using this hybrid Legendre polynomials approximation method. Moreover, the Hermite differential equation of order zero is solved by using hybrid Legendre polynomials approximation method to explain the importance and the application of the technique of this method.

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13

Al-Refai, Mohammed, Muhammed Syam, and Qasem Al-Mdallal. "On the Fractional Legendre Equation and Fractional Legendre Functions." Progress in Fractional Differentiation and Applications 3, no. 2 (April 1, 2017): 93–102. http://dx.doi.org/10.18576/pfda/030202.

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14

Salem, L. D., and H. S. Wio. "Some integrals involving Legendre polynomials and associated Legendre functions." Journal of Physics A: Mathematical and General 22, no. 20 (October 21, 1989): 4331–38. http://dx.doi.org/10.1088/0305-4470/22/20/012.

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15

Putri, Nuraini Saura, and Oktavian Aditya Nugraha. "PERBANDINGAN STRUKTUR, FUNGSI, DAN NILAI BUDAYA PADA LEGENDA TELAGA NGEBEL PONOROGO DAN LEGENDA DANAU RANU PASURUAN." Jurnal Pena Indonesia 3, no. 2 (October 30, 2017): 201. http://dx.doi.org/10.26740/jpi.v3n2.p201-222.

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Efforts to preserve the legendary cultural heritage and in the framework of seeking the identity of the nation as well as the diversity of ethnic groups in Indonesia, one way to do is to collect and care for folklore. This research as a means to treat foklor legend of Lake Ranu and Ngebel legend Telaga. Both legends have almost the same story, both the character and the story. The purpose of this study is to determine the similarities and differences in the two legends, through its structure, function and cultural values. Such long-term goal in this study is, to provide knowledge about oral literature research on literature students who can be used as a reference for the future. In addition, this research can also provide new knowledge for the general public about the storyline of the legend of Lake Ngebel and the legend of Lake Ranu. Based on the results of research that has been done then can be drawn a conclusion, namely: the structure there are four equations and has three differences. Something contained in the two legends have three functions, namely; as entertainment, as a means of legitimizing institutions and cultural institutions, as child educators. For the cultural values contained in these two legends are the value of Diktatik, Ethical, and Religious values

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16

Durand, Loyal. "Fractional operators and special functions. II. Legendre functions." Journal of Mathematical Physics 44, no. 5 (May 2003): 2266–92. http://dx.doi.org/10.1063/1.1561594.

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17

Mandal, Nanigopal, and B. N. Mandal. "Expansion of a class of functions into an integral involving associated Legendre functions." International Journal of Mathematics and Mathematical Sciences 17, no. 2 (1994): 293–300. http://dx.doi.org/10.1155/s0161171294000438.

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A theorem for expansion of a class of functions into an integral involving associated Legendre functions is obtained in this paper. This is a somewhat general integral expansion formula for a functionf(x)defined in(x1,x2)where-1<x1<x2<1, which is perhaps useful in solving certain boundary value problems of mathematical physics and of elasticity involving conical boundaries.

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18

Winch, D. E., and P. H. Roberts. "Derivatives of addition theorems for Legendre functions." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 37, no. 2 (October 1995): 212–34. http://dx.doi.org/10.1017/s0334270000007670.

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AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

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19

Virchenko, N. A. "Applications of Legendre's generalized associated functions." Ukrainian Mathematical Journal 39, no. 2 (1987): 124–30. http://dx.doi.org/10.1007/bf01057490.

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20

Gervois, A., and H. Navelet. "Integrals of three Bessel functions and Legendre functions. I." Journal of Mathematical Physics 26, no. 4 (April 1985): 633–44. http://dx.doi.org/10.1063/1.526600.

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21

Love, E. R., and M. N. Hunter. "Expansions in Series of Legendre Functions." Proceedings of the London Mathematical Society s3-64, no. 3 (May 1992): 579–601. http://dx.doi.org/10.1112/plms/s3-64.3.579.

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22

Hall, Chris. "L-functions of twisted Legendre curves." Journal of Number Theory 119, no. 1 (July 2006): 128–47. http://dx.doi.org/10.1016/j.jnt.2005.10.004.

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23

Bakaleinikov, Leonid, and Alexander Silbergleit. "Uniform asymptotic expansion of Legendre functions." Journal of Mathematical Physics 61, no. 8 (August 1, 2020): 083503. http://dx.doi.org/10.1063/5.0016206.

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24

Lohöfer, G. "Inequalities for the Associated Legendre Functions." Journal of Approximation Theory 95, no. 2 (November 1998): 178–93. http://dx.doi.org/10.1006/jath.1998.3207.

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25

Thylwe, Karl-Erik, and Patrick McCabe. "On Calculations of Legendre Functions and Associated Legendre Functions of the First Kind of Complex Degree." Communications in Theoretical Physics 64, no. 1 (July 1, 2015): 9–12. http://dx.doi.org/10.1088/0253-6102/64/1/9.

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26

Malits, P. "On certain Heun functions, associated functions of discrete variables, and applications." International Journal of Mathematics and Mathematical Sciences 2003, no. 59 (2003): 3717–51. http://dx.doi.org/10.1155/s0161171203211479.

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Two sets of the Heun functions are introduced via integrals. Theorems about expanding functions with respect to these sets are proven. A number of integral and series representations as well as integral equations and asymptotic formulas are obtained for these functions. Some of the coefficients of the series are orthogonal (J-orthogonal) functions of discrete variables and may be interpreted as orthogonal polynomials. Other sets of the coefficients are biorthonormal. Expanding infinite vectors to series with respect to the coefficients is discussed. Certain Legendre functions of complex degree are limiting cases of the studied functions. This leads to new relations for Legendre functions and associated integral transforms. The treated Heun functions find a use for solving dual Fourier series equations which are reduced to the Fredholm integral equations of the second kind. Explicit solutions are obtained in a special case.

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27

Qashash, Gamal Ali. "Generating Functions for Legendre Polynomials by using Group Theoretic Method." University of Aden Journal of Natural and Applied Sciences 23, no. 1 (April 30, 2019): 173–80. http://dx.doi.org/10.47372/uajnas.2019.n1.a14.

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In this paper we obtain generating functions for the Legendre polynomials \(P_n (x)\) in its modified form by using Weisner's group theoretic method. Whereas, we deployed it to determine the new generating relations between the generalized Legendre polynomials and with easy way. The ideas in consent with proofs are originated from the book of McBride [8] and is used to determine new generating relations which involve modified Legendre polynomials.

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28

Gervois, A., and H. Navelet. "Integrals of some three Bessel functions and Legendre functions. II." Journal of Mathematical Physics 26, no. 4 (April 1985): 645–55. http://dx.doi.org/10.1063/1.526601.

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29

Maier, Robert S. "Legendre functions of fractional degree: transformations and evaluations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2188 (April 2016): 20160097. http://dx.doi.org/10.1098/rspa.2016.0097.

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Associated Legendre functions of fractional degree appear in the solution of boundary value problems on wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer, they can be expressed using complete elliptic integrals. In this study, many transformations are derived, which reduce the case when the degree differs from an integer by one-third, one-fourth or one-sixth to the classical case. These transformations or identities facilitate the symbolic manipulation and evaluation of Legendre and Ferrers functions. They generalize both Ramanujan's transformations of elliptic integrals and Whipple's formula, which relates Legendre functions of the first and second kinds. The proofs employ algebraic coordinate transformations, specified by algebraic curves.

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30

Wang, Jian Qiang, Hao Yuan Chen, and Yin Fu Chen. "The Analysis of the Associated Legendre Functions with Non-Integral Degree." Applied Mechanics and Materials 130-134 (October 2011): 3001–5. http://dx.doi.org/10.4028/www.scientific.net/amm.130-134.3001.

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Spherical cap harmonic (SCH) theory has been widely used to format regional model of fields that can be expressed as the gradient of a scalar potential. The functions of this method consist of trigonometric functions and associated Legendre functions with integral-order but non-integral degree. Evidently, the constructing and computing of Legendre functions are the core content of the spherical cap functions. In this paper,the approximated calculation method of the normalized association Legendre functions with non-integral degree is introduced and an analysis of the entire order of associated non-Legendre function calculation is presented. Besides, we use the Muller method to search out for all intrinsic values. The results showed that the highest order of spherical harmonic function for constructing regional model of fields is limited, thus high-resolution spherical harmonic structure of local gravity field need to be improved.

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31

Novikova, Elena, and Alexander Dmitrenko. "Problems and methods of calculating the Legendre functions of arbitrary degree and order." Geodesy and Cartography 65, no. 2 (December 1, 2016): 283–312. http://dx.doi.org/10.1515/geocart-2016-0017.

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Abstract The known standard recursion methods of computing the full normalized associated Legendre functions do not give the necessary precision due to application of IEEE754-2008 standard, that creates a problems of underflow and overflow. The analysis of the problems of the calculation of the Legendre functions shows that the problem underflow is not dangerous by itself. The main problem that generates the gross errors in its calculations is the problem named the effect of “absolute zero”. Once appeared in a forward column recursion, “absolute zero” converts to zero all values which are multiplied by it, regardless of whether a zero result of multiplication is real or not. Three methods of calculating of the Legendre functions, that removed the effect of “absolute zero” from the calculations are discussed here. These methods are also of interest because they almost have no limit for the maximum degree of Legendre functions. It is shown that the numerical accuracy of these three methods is the same. But, the CPU calculation time of the Legendre functions with Fukushima method is minimal. Therefore, the Fukushima method is the best. Its main advantage is computational speed which is an important factor in calculation of such large amount of the Legendre functions as 2 401 336 for EGM2008.

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32

Setiawan, Ari. "Visualisasi Fungsi Legendre dan Perilaku dari Permukaan Bola Harmonik dengan Menggunakan Bahasa Pemrograman Matlab." Jurnal Fisika Indonesia 22, no. 1 (March 12, 2020): 1. http://dx.doi.org/10.22146/jfi.v22i1.43533.

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Telah dilakukan penelitian yang berhubungan dengan medan potensial gravitasi di permukaan bumi yang merupakan persamaan harmonik bola. Harmonik bola tersebut merupakan penyelesaian dari persamaan Laplace, dalam sistem koordinat bola. Persamaan ini dikenal dengan Associated Legendre Functions. Pada penelitian ini dikembangkan visualisasi Associated Legendre Functions atau disebut juga dengan Spherical Harmonic Functions dalam tampilan 1D, 2D, 3D dengan menggunakan pemrograman MATLAB. Telah berhasil dikembang visualisasi dalam koordinat bola untuk fungsi r + Pnm(x) dengan r = 5 dan Pnm(x) yaitu Associated Legendre Functions untuk derajat n = dari 1 sampai 7 dan dengan orde m ≤ n.

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33

Wang, Houjun, John P. Boyd, and Rashid A. Akmaev. "On computation of Hough functions." Geoscientific Model Development 9, no. 4 (April 20, 2016): 1477–88. http://dx.doi.org/10.5194/gmd-9-1477-2016.

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Abstract. Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the commonly used unnormalized associated Legendre polynomial expansion method. Comparable results are obtained using both methods. For the first method we note some details on numerical implementation. The Chebyshev collocation method was first used for the Laplace tidal problem by Boyd (1976) and is relatively easy to use. A compact MATLAB code is provided for this method. We also illustrate the importance and effect of including a parity factor in Chebyshev polynomial expansions for modes with odd zonal wave numbers.

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34

Singh, B. M., J. Rokne, and R. S. Dhaliwal. "The Study of Triple Integral Equations with Generalized Legendre Functions." Abstract and Applied Analysis 2008 (2008): 1–12. http://dx.doi.org/10.1155/2008/395257.

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A method is developed for solutions of two sets of triple integral equations involving associated Legendre functions of imaginary arguments. The solution of each set of triple integral equations involving associated Legendre functions is reduced to a Fredholm integral equation of the second kind which can be solved numerically.

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35

Kalla, S. L., N. Virchenko, and O. Lisetska. "Some results involving generalized associated Legendre functions." Integral Transforms and Special Functions 23, no. 2 (February 2012): 105–14. http://dx.doi.org/10.1080/10652469.2011.564379.

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36

Babich, V. M., and K. D. Cherednichenko. "On fock's type asymptotics of legendre functions." Integral Transforms and Special Functions 5, no. 1-2 (May 1997): 1–18. http://dx.doi.org/10.1080/10652469708819123.

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37

Perez, H., and S. Tsujii. "IIR adaptive filtering via discrete Legendre functions." Electronics Letters 24, no. 8 (1988): 450. http://dx.doi.org/10.1049/el:19880304.

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38

Szalay, V. "The overlap integral of associated Legendre functions." Journal of Physics A: Mathematical and General 23, no. 12 (June 21, 1990): 2689–94. http://dx.doi.org/10.1088/0305-4470/23/12/041.

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39

Kalnishkan, Yuri, Volodya Vovk, and Michael V. Vyugin. "Loss functions, complexities, and the Legendre transformation." Theoretical Computer Science 313, no. 2 (February 2004): 195–207. http://dx.doi.org/10.1016/j.tcs.2003.11.005.

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40

Földváry, Lóránt. "Sine series expansion of associated Legendre functions." Acta Geodaetica et Geophysica 50, no. 2 (January 13, 2015): 243–59. http://dx.doi.org/10.1007/s40328-014-0092-2.

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41

Méndez-Pérez, J. M. R., and G. Miquel Morales. "The Finite Legendre Transformation of Generalized Functions." Rocky Mountain Journal of Mathematics 28, no. 4 (December 1998): 1371–89. http://dx.doi.org/10.1216/rmjm/1181071722.

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42

Rahman, Mizan, and Qazi M. Tariq. "Addition formulas for $q$-Legendre-type functions." Methods and Applications of Analysis 6, no. 1 (1999): 3–20. http://dx.doi.org/10.4310/maa.1999.v6.n1.a1.

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43

Haggard, Paul W. "On Legendre numbers of the second kind." International Journal of Mathematics and Mathematical Sciences 11, no. 4 (1988): 815–22. http://dx.doi.org/10.1155/s0161171288000997.

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The Legendre numbers of the second kind, an infinite set of rational numbers, are defined from the associated Legendre functions. An explicit formula and a partial table for these numbers are given and many elementary properties are presented. A connection is shown between Legendre numbers of the first and second kinds. Extended Legendre numbers of the first and second kind are defined in a natural way and these are expressed in terms of those of the second and first kind, respectively. Two other sets of rational numbers are defined from the associated Legendre functions by taking derivatives and evaluating these atx=0. One of these sets is connected to Legendre numbers of the first find while the other is connected to Legendre numbers of the second kind. Some series are also discussed.

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44

Haggard, Paul W. "On Legendre numbers." International Journal of Mathematics and Mathematical Sciences 8, no. 2 (1985): 407–11. http://dx.doi.org/10.1155/s0161171285000436.

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The Legendre numbers, an infinite set of rational numbers are defined from the associated Legendre functions and several elementary properties are presented. A general formula for the Legendre numbers is given. Applications include summing certain series of Legendre numbers and evaluating certain integrals. Legendre numbers are used to obtain the derivatives of all orders of the Legendre polynomials atx=1.

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45

Haggard, Paul W. "Some applications of Legendre numbers." International Journal of Mathematics and Mathematical Sciences 11, no. 2 (1988): 405–12. http://dx.doi.org/10.1155/s0161171288000481.

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The associated Legendre functions are defined using the Legendre numbers. From these the associated Legendre polynomials are obtained and the derivatives of these polynomials atx=0are derived by using properties of the Legendre numbers. These derivatives are then used to expand the associated Legendre polynomials andxnin series of Legendre polynomials. Other applications include evaluating certain integrals, expressing polynomials as linear combinations of Legendre polynomials, and expressing linear combinations of Legendre polynomials as polynomials. A connection between Legendre and Pascal numbers is also given.

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46

Haq, Sirazul, Maggie Aphane, Mohammad Khan, and Nicola Fabiano. "Certain integrals involving generalized Mittag-Leffler type functions." Vojnotehnicki glasnik 70, no. 4 (2022): 797–817. http://dx.doi.org/10.5937/vojtehg70-40296.

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Introduction/purpose: Certain integrals involving the generalized Mittag-Leffler function with different types of polynomials are established. Methods: The properties of the generalized Mittag-Leffler function are used in conjunction with different kinds of polynomials such as Jacobi, Legendre, and Hermite in order to evaluate their integrals. Results: Some integral formulae involving the Legendre function, the Bessel Maitland function and the generalized hypergeometric functions are derived. Conclusions: The results obtained here are general in nature and could be useful to establish further integral formulae involving other kinds of polynomials.

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Sablotny, Antje. "“zu grob gewest”: Metainvective Communication in Confessional Disputes over Narration of the Saints in the Sixteenth Century." Journal of Early Modern Christianity 10, no. 1 (April 1, 2023): 143–65. http://dx.doi.org/10.1515/jemc-2023-2042.

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Abstract The article is devoted to coarse uses of language as a subject of dispute in confessional controversies over legendary narration. Such metainvective forms of communication are systematized, and questioned with regard to their functions: in the Protestant Lügenden (word combination of “legend” and “lie,” lying legends) and their Catholic replies, the “true” faith and its defense are connected with communicative behavior. Whereas Lutherans are above all effective at adopting coarse speech and the metainvective reproach of lying, the Counter-Reformation argumentation develops the strategic potential of metainvective communication in very different ways. Metainvective statements become a weapon particularly when they are absorbed into figures of meta-metainvective, which not only display the coarse speech but reveal and then criticize the strategy behind it. The tension between polemical prefaces and annotated miracle narratives in the Lügenden as well as the thematic proliferation of the legend discussion in the Catholic reports and sermons are ultimately shown to be genre-dynamic effects of the use of metainvectives.

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Wang, Tian-Jun, Chao Zhang, and Qiong Zhang. "Mixed Spectral Method for Heat Transfer Using Generalised Hermite Functions and Legendre Polynomials." East Asian Journal on Applied Mathematics 6, no. 4 (October 19, 2016): 448–65. http://dx.doi.org/10.4208/eajam.231015.240816a.

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AbstractWe propose amixed spectral method for heat transfer in unbounded domains, using generalised Hermite functions and Legendre polynomials. Some basic results on the mixed generalised Hermite-Legendre orthogonal approximation are established, which plays important roles in spectral methods for various problems defined on unbounded domains. As an example, the mixed generalised Hermite-Legendre spectral scheme is constructed for anisotropic heat transfer. Its convergence is proven, and some numerical results demonstrate the spectral accuracy of this approach.

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Karatsuba, E. A., and M. Vuorinen. "On Hypergeometric Functions and Generalizations of Legendre's Relation." Journal of Mathematical Analysis and Applications 260, no. 2 (August 2001): 623–40. http://dx.doi.org/10.1006/jmaa.2001.7499.

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Putri, Eka, Mulyadi Eko Purnomo, and Santi Oktarina. "SOCIAL CONFLICTS FOUND IN THE LEGENDS OF DAYANG MERINDU AND NAYA SENTIKA AND THEIR IMPLEMENTATIONS IN LITERATURE LEARNING." English Review: Journal of English Education 11, no. 1 (February 28, 2023): 109–22. http://dx.doi.org/10.25134/erjee.v11i1.7594.

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This research describes the realizations of social conflicts, the function of social conflict, the similarities and differences of Dayang Merindu and Naya Sentika legends, and the implications on literature learning. The research applied a descriptive comparative method supported by an objective approach. Then, the researchers analyzed the data by collecting, reducing, presenting, and concluding the data. The data sources were the legends of Dayang Merindu and Naya Sentika. The research data were social conflict realizations and functions of the legends in the form of text excerpts. The results showed differences in the legends. The legend of Dayang Merindu had three types of conflicts: interpersonal, interest, and inter-social conflicts. The legend of Naya Sentika had four types of conflicts: interpersonal, interest, inter-group, and international conflicts. The similarities of the legends were the social functions of the conflicts. They were two functions: solidarity and communicative functions. The current research results are applicable for literature learning and further reading sources as the realization of literature appreciation.

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