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Journal articles on the topic 'Marchaud derivative'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

PENG, WEN LIANG, KUI YAO, XIA ZHANG, and JIA YAO. "BOX DIMENSION OF WEYL–MARCHAUD FRACTIONAL DERIVATIVE OF LINEAR FRACTAL INTERPOLATION FUNCTIONS." Fractals 27, no. 04 (June 2019): 1950058. http://dx.doi.org/10.1142/s0218348x19500580.

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This paper mainly explores Weyl–Marchaud fractional derivative of linear fractal interpolation functions (FIFs). We prove that Weyl–Marchaud fractional derivative of a linear FIF is still a linear FIFs. More generally, we get a conclusion that there exists some linear relationship between the order of Weyl–Marchaud fractional derivative and box dimension of linear FIFs.

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2

Babenko, V. F., and M. S. Churilova. "On inequalities of Kolmogorov type for fractional derivatives of functions defined on the real domain." Researches in Mathematics 16 (February 7, 2021): 28. http://dx.doi.org/10.15421/240804.

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We obtain new inequalities that generalize known result of Geisberg, which was obtained for fractional Marchaud derivatives, to the case of higher derivatives, at that the fractional derivative is a Riesz one. The inequality with second higher derivative is sharp.

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3

Mamatov, Tulkin, Nemat Mustafoev, Dilshod Barakaev, and Rano Sabirova. "Hardy-Littlewood-Type Theorem for Mixed Fractional Integrals in Hölder Spaces." Indian Journal of Advanced Mathematics 1, no. 2 (October 10, 2021): 15–19. http://dx.doi.org/10.54105/ijam.b1105.101221.

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We study mixed Riemann-Liouville fractional integration operators and mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The obtained are results generalized to the case of Hölder spaces with power weight. Keywords: functions of two variables, fractional derivative of Marchaud form, mixed fractional derivative, weight, mixed fractional integral, Hölder space.

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4

LIANG, Y. S., and N. LIU. "FRACTAL DIMENSIONS OF WEYL–MARCHAUD FRACTIONAL DERIVATIVE OF CERTAIN ONE-DIMENSIONAL FUNCTIONS." Fractals 27, no. 07 (November 2019): 1950114. http://dx.doi.org/10.1142/s0218348x19501147.

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Fractal dimensions of Weyl–Marchaud fractional derivative of certain continuous functions are investigated in this paper. Upper Box dimension of Weyl–Marchaud fractional derivative of certain continuous functions with Box dimension one has been proved to be no more than the sum of one and its order.

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5

Djida, Jean-Daniel, and Arran Fernandez. "Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions." Axioms 7, no. 3 (September 1, 2018): 65. http://dx.doi.org/10.3390/axioms7030065.

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The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative.

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6

Parfinovych, N. V., and V. V. Pylypenko. "Kolmogorov inequalities for norms of Marchaud-type fractional derivatives of multivariate functions." Researches in Mathematics 28, no. 2 (December 28, 2020): 10. http://dx.doi.org/10.15421/242007.

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7

Babenko, V. F., and T. V. Matveeva. "Inequalities of Kolmogorov type for fractional derivatives of multivariable functions." Researches in Mathematics 16 (February 7, 2021): 3. http://dx.doi.org/10.15421/240801.

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8

Mamatov, Tulkin, Nemat Mustafoev, Dilshod Barakaev, and Rano Sabirova. "Hardy-Littlewood-Type Theorem for Mixed Fractional Integrals in Hölder Spaces." Indian Journal of Advanced Mathematics 1, no. 2 (October 10, 2021): 15–19. http://dx.doi.org/10.35940/ijam.b1105.101221.

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We study mixed Riemann-Liouville fractional integration operators and mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The obtained are results generalized to the case of Hölder spaces with power weight.

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9

Kumar, Pushpendra, and Muzammil Khan. "Charbonnier-Marchaud Based Fractional Variational Model for Motion Estimation in Multispectral Vision System." Journal of Physics: Conference Series 2327, no. 1 (August 1, 2022): 012031. http://dx.doi.org/10.1088/1742-6596/2327/1/012031.

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Abstract As we are aware that motion estimation is an active and challenging area of vision system, which leads to the applications of computer vision. In general, motion detection and tracking in the image sequence (video) is carried out based on optical flow. In the recent-past, researchers have made a significant contribution to the estimation of optical flow through integer order-based variational models, but these are limited to integer order differentiation. In this paper, a nonlinear modeling of fractional order variational model in optical flow estimation is introduced using the Charbonnier norm, which can be scaled to integer order L 1-norm. In particular, the variational functional is formulated by the inclusion of a non-quadratic penalty term, regularization term and the Marchaud’s fractional derivative. This non-quadratic penalty provides effective robustness against outliers, whereas the Marchaud’s fractional derivative possesses a non-local character, and therefore is capable to deal with discontinuous information about texture and edges, and yields a dense flow field. The numerical discretization of the Marchaud’s fractional derivative is employed with the help of Grünwald–Letnikov fractional derivative. The resulting nonlinear system is converted into a linear system and solved by an efficient numerical technique. Several experiments are performed over a spectrum of datasets. The robustness and accuracy of the proposed model are shown under different amounts of noise and through detailed comparisons with the recently published works.

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10

Kilbas, Anatoly, and Anatoly Titioura. "NONLINEAR DIFFERENTIAL EQUATIONS WITH MARCHAUD‐HADAMARD-TYPE FRACTIONAL DERIVATIVE IN THE WEIGHTED SPACE OF SUMMABLE FUNCTIONS." Mathematical Modelling and Analysis 12, no. 3 (September 30, 2007): 343–56. http://dx.doi.org/10.3846/1392-6292.2007.12.343-356.

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The paper is devoted to the study of the Cauchy‐type problem for the nonlinear differential equation of fractional order 0 < α < 1: containing the Marchaud-Hadamard-type fractional derivative (Dα 0+, μ y)(x), on the half-axis R+ = (0, +oo) in the space Xp,α c,0 (R+) defined for α > 0 by where Xp c, 0 (R+) is the subspace of Xp c (R+) of functions g Xp c (R + ) with compact support on infinity: g(x) = 0 for large enough x > R. The equivalence of this problem and of the nonlinear Volterra integral equation is established. The existence and uniqueness of the solution y(x) of the above Cauchy‐type problem is proved by using the Banach fixed point theorem. Solution in closed form of the above problem for the linear differential equation with f[x, y(x)] = λy(x) + f(x) is constructed. The corresponding assertions for the differential equations with the Marchaud‐Hadamard fractional derivative (Dα 0+ y)(x) are presented. Examples are given.

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11

Gulgowski, Jacek, Tomasz P. Stefański, and Damian Trofimowicz. "On Applications of Elements Modelled by Fractional Derivatives in Circuit Theory." Energies 13, no. 21 (November 4, 2020): 5768. http://dx.doi.org/10.3390/en13215768.

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In this paper, concepts of fractional-order (FO) derivatives are reviewed and discussed with regard to element models applied in the circuit theory. The properties of FO derivatives required for the circuit-level modeling are formulated. Potential problems related to the generalization of transmission-line equations with the use of FO derivatives are presented. It is demonstrated that some formulations of FO derivatives have limited applicability in the circuit theory. Out of the most popular approaches considered in this paper, only the Grünwald–Letnikov and Marchaud definitions (which are actually equivalent) satisfy the semigroup property and are naturally representable in the phasor domain. The generalization of this concept, i.e., the two-sided fractional Ortigueira–Machado derivative, satisfies the semigroup property, but its phasor representation is less natural. Other ideas (including the Riemann–Liouville and Caputo derivatives—with a finite or an infinite base point) seem to have limited applicability.

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12

Zhang, Qi, and Yongshun Liang. "The Weyl–Marchaud fractional derivative of a type of self-affine functions." Applied Mathematics and Computation 218, no. 17 (May 2012): 8695–701. http://dx.doi.org/10.1016/j.amc.2012.01.077.

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13

Liang, Y., K. Yao, and W. Xiao. "The fractal dimensions of graphs of the Weierstrass function with the Weyl-Marchaud fractional derivative." Journal of Physics: Conference Series 96 (February 1, 2008): 012111. http://dx.doi.org/10.1088/1742-6596/96/1/012111.

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14

Kui, Yao, Weiyi Su, and Yongshun Liang. "The upper bound of box dimension of the Weyl-Marchaud derivative of self-affine curves." Analysis in Theory and Applications 26, no. 3 (September 2010): 222–27. http://dx.doi.org/10.1007/s10496-010-0222-9.

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15

Yao, K., Y. S. Liang, and J. X. Fang. "The fractal dimensions of graphs of the Weyl-Marchaud fractional derivative of the Weierstrass-type function." Chaos, Solitons & Fractals 35, no. 1 (January 2008): 106–15. http://dx.doi.org/10.1016/j.chaos.2007.04.017.

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16

Kukushkin, M. V. "ON SOME QUALITATIVE PROPERTIES OF THE OPERATOR OF FRACTIONAL DIFFERENTIATION IN KIPRIYANOV SENSE." Vestnik of Samara University. Natural Science Series 23, no. 2 (September 21, 2017): 32–43. http://dx.doi.org/10.18287/2541-7525-2017-23-2-32-43.

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In this paper we investigated the qualitative properties of the operator of fractional differentiation in Kipriyanov sense. Based on the concept of multidimensional generalization of operator of fractional differentiation in Marchaud sense we have adapted earlier known techniques of proof theorems of one-dimensional theory of fractional calculus for the operator of fractional differentiation in Kipriyanov sense. Along with the previously known definition of the fractional derivative in the direction we used a new definition of multidimensional fractional integral in the direction of allowing you to expand the domain of definition of formally adjoint operator. A number of theorems that have analogs in one-dimensional theory of fractional calculus is proved. In particular the sufficient conditions of representability of a fractional integral in the direction are received. Integral equality the result of which is the construction of the formal adjoint operator defined on the set of functions representable by the fractional integral in direction is proved.

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17

Ferrari, Fausto. "Weyl and Marchaud Derivatives: A Forgotten History." Mathematics 6, no. 1 (January 3, 2018): 6. http://dx.doi.org/10.3390/math6010006.

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18

Almeida, Ricardo, and Delfim F. M. Torres. "An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order." Scientific World Journal 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/915437.

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We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.

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19

Babenko, V. F., and M. S. Churilova. "On inequalities for $L_p$-norms of fractional derivatives on the real domain." Researches in Mathematics 15 (February 15, 2021): 26. http://dx.doi.org/10.15421/240704.

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We obtain new inequalities for fractional Marchaud derivatives of functions defined on the whole real domain, in integral metric ($1 \leqslant p < \infty$); for $p = 1$ we establish the sharpness of obtained inequalities.

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20

Rogosin, Sergei, and Maryna Dubatovskaya. "Letnikov vs. Marchaud: A Survey on Two Prominent Constructions of Fractional Derivatives." Mathematics 6, no. 1 (December 25, 2017): 3. http://dx.doi.org/10.3390/math6010003.

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21

Lyakhov, L. N., and E. L. Sanina. "Schlömilch polynomials: Riesz’s interpolation formula for B-derivatives and Bernstein’s inequality for Weyl-Marchaud fractional B-derivatives." Doklady Mathematics 76, no. 3 (December 2007): 916–20. http://dx.doi.org/10.1134/s1064562407060270.

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22

Yakhshiboev, M. U. "Hadamard-type fractional integrals and Marchaud-Hadamard-type fractional derivatives in the spaces with power weight." Uzbek Mathematical Journal 2019, no. 3 (September 17, 2019): 155–74. http://dx.doi.org/10.29229/uzmj.2019-3-17.

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23

Babenko, Vladislav F., Mariya S. Churilova, Nataliia V. Parfinovych, and Dmytro S. Skorokhodov. "Kolmogorov type inequalities for the Marchaud fractional derivatives on the real line and the half-line." Journal of Inequalities and Applications 2014, no. 1 (2014): 504. http://dx.doi.org/10.1186/1029-242x-2014-504.

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24

Efthimiou, Christoforos. "The Concept of Political Difference in Oliver Marchart and its Relationship with the Heideggerian Concept of Ontological Difference." Conatus 4, no. 1 (October 31, 2019): 61. http://dx.doi.org/10.12681/cjp.18863.

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The concept of political difference concerns the distinction between politics and the political. The political refers to the ontological making possible of the different domains of society, including the domain of politics in the narrow sense. Political difference was introduced as a reaction to the theoretical controversy between foundationalism and anti-foundationalism. This reaction took the form of post-foundationalism. According to Marchart, post-foundationalism does not entirely deny the possibility of grounding. It denies only the possibility of an ultimate transcendent foundation insofar as this ontological impossibility makes possible the historical and contingent grounds in plural.The Heideggerian concept of ontological difference also undermines the possibility of an ultimate ontical ground which establishes the presence of all the other beings. If one wants to think beyond the concept of ground, one should obtain a clear understanding of Being as Being, namely one should grasp the Being in its difference from beings. All the same, Heidegger tends to replace the ontical grounds of metaphysics with Being itself as a new kind of ultimate ontological foundation.On the other hand, one can detect in many points of Heideggerian argumentation traces of a second alternative understanding of ontological difference which does not belong in Heidegger’s intentions and which undermines the primordiality of Being. This alternative understanding establishes a reciprocity between Being and beings. In our view, political difference not only is based in this second way of understanding but, at the same time, develops more decisively the mutual interdependence between Being and beings.In political difference the grounding part, namely the political, possesses both a grounding character and a derivative one. Politics and political both grounds and dislocate each other in an incessant and oscillating, historical procedure which undermines any form of completion of the social.

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25

Divo, Eduardo, and Alain J. Kassab. "An Efficient Localized Radial Basis Function Meshless Method for Fluid Flow and Conjugate Heat Transfer." Journal of Heat Transfer 129, no. 2 (May 25, 2006): 124–36. http://dx.doi.org/10.1115/1.2402181.

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A localized radial basis function (RBF) meshless method is developed for coupled viscous fluid flow and convective heat transfer problems. The method is based on new localized radial-basis function (RBF) expansions using Hardy Multiquadrics for the sought-after unknowns. An efficient set of formulae are derived to compute the RBF interpolation in terms of vector products thus providing a substantial computational savings over traditional meshless methods. Moreover, the approach developed in this paper is applicable to explicit or implicit time marching schemes as well as steady-state iterative methods. We apply the method to viscous fluid flow and conjugate heat transfer (CHT) modeling. The incompressible Navier–Stokes are time marched using a Helmholtz potential decomposition for the velocity field. When CHT is considered, the same RBF expansion is used to solve the heat conduction problem in the solid regions enforcing temperature and heat flux continuity of the solid/fluid interfaces. The computation is accelerated by distributing the load over several processors via a domain decomposition along with an interface interpolation tailored to pass information through each of the domain interfaces to ensure conservation of field variables and derivatives. Numerical results are presented for several cases including channel flow, flow in a channel with a square step obstruction, and a jet flow into a square cavity. Results are compared with commercial computational fluid dynamics code predictions. The proposed localized meshless method approach is shown to produce accurate results while requiring a much-reduced effort in problem preparation in comparison to other traditional numerical methods.

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26

Trigolo Pahim, Adriano, Maria Fernanda Pereira Gomes, and Lislaine Aparecida Fracolli. "Estratégia saúde da família - a ótica dos cuidadores de crianças." Revista de Enfermagem UFPE on line 12, no. 3 (March 3, 2018): 607. http://dx.doi.org/10.5205/1981-8963-v12i3a24120p607-617-2018.

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RESUMOObjetivo: avaliar sob a ótica dos cuidadores de crianças menores de dois anos e usuários adultos se os atributos essenciais e derivados da atenção primária à saúde estavam presentes e como estavam sendo operacionalizados na Estratégia Saúde da Família. Método: estudo quantitativo, descritivo e transversal realizado em três municípios da região de Presidente Prudente, Estado de São Paulo, Brasil. Utilizou-se o instrumento de avaliação da atenção primária à saúde (PCATool) na versão criança e adulto, em que 176 cuidadores de crianças menores de dois anos e 181 adultos usuários participaram da pesquisa. Resultados: os atributos da atenção primária à saúde estavam presentes na Estratégia Saúde da Família da região; porém, alguns atributos como Coordenação – Integração de Cuidados, Acesso de Primeiro Contato – Acessibilidade e Orientação Familiar precisavam melhorar sua operacionalização. Conclusão: os atributos da atenção primária à saúde devem estar adequadamente incorporados à prática dos profissionais da saúde por meio da capacitação e valorização das ações que caracterizam a Estratégia Saúde da Família como modelo com foco na família. Descritores: Avaliação em Saúde; Estratégia Saúde da Família; Atenção Primária à Saúde; Saúde Pública; Saúde da Criança; Saúde do Adulto. ABSTRACT Objective: to assess whether the essential attributes and derivatives of primary health care were present and how they were being implemented in the Family Health Strategy from the point of view of caregivers of children under two years of age and adult users. Method: quantitative, descriptive and cross-sectional study conducted in three municipalities of the region of Presidente Prudente, State of São Paulo, Brazil. We used the child and adult versions of the Primary Care Assessment Tool (PCATool), and 176 caregivers of children under two years of age and 181 adult users participated in the research. Results: the attributes of primary health care were present in the Family Health Strategy in the region; however, some attributes, such as healthcare coordination-integration, access to initial contact–accessibility, and family guidance needed to have their implementation improved. Conclusion: the attributes of primary health care should be properly incorporated into the practice of health professionals through training and actions that characterize the Family Health Strategy as a model with a focus on the family. Descritoprs: Health Assessment; Family Health Strategy; Primary Health Care; Public Health; Child Health; Adult Health. RESUMEN Objetivo: evaluar bajo la óptica de los cuidadores de niños menores de dos años y adultos usuarios si los atributos esenciales y derivados de la atención primaria de salud estaban presentes y cómo estaban siendo puestos en marcha en la Estrategia Salud de la Familia. Método: estudio cuantitativo, descriptivo y transversal realizado en tres municipios de la región de Presidente Prudente, Estado de São Paulo, Brasil. Se usaron las versiones niño y adulto del Instrumento de Evaluación de la Atención Primaria (PCATool) y 176 cuidadores de niños menores de dos años y 181 adultos participaron en la investigación. Resultados: los atributos de la atención primaria de salud estaban presentes en la Estrategia Salud de la Familia en la región; sin embargo, algunos atributos tales como coordinación-integración de los cuidados, acceso a primer contacto–accesibilidad y asesoramiento familiar necesitaban mejorar su puesta en marcha. Conclusión: los atributos de la atención primaria de salud deben ser debidamente incorporados a la práctica de profesionales de la salud a través de capacitación y desarrollo de acciones que caracterizan la Estrategia Salud de la Familia como un modelo con enfoque en la familia. Descriptores: Evaluación De La Salud; Estrategia Salud de la Familia; Atención Primaria de Salud; Salud Pública; Salud Del Niño; Salud del Adulto.

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27

CHANDRA, SUBHASH, and SYED ABBAS. "ANALYSIS OF MIXED WEYL–MARCHAUD FRACTIONAL DERIVATIVE AND BOX DIMENSIONS." Fractals, July 28, 2021, 2150145. http://dx.doi.org/10.1142/s0218348x21501450.

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The calculus of the mixed Weyl–Marchaud fractional derivative has been investigated in this paper. We prove that the mixed Weyl–Marchaud fractional derivative of bivariate fractal interpolation functions (FIFs) is still bivariate FIFs. It is proved that the upper box dimension of the mixed Weyl–Marchaud fractional derivative having fractional order [Formula: see text] of a continuous function which satisfies [Formula: see text]-Hölder condition is no more than [Formula: see text] when [Formula: see text], [Formula: see text], [Formula: see text], which reveals an important phenomenon about linearly increasing effect of dimension of the mixed Weyl–Marchaud fractional derivative. Furthermore, we deduce box dimension of the graph of the mixed Weyl–Marchaud fractional derivative of a continuous function which is defined on a rectangular region in [Formula: see text] and also, we analyze that the mixed Weyl–Marchaud fractional derivative of a function preserves some basic properties such as continuity, bounded variation and boundedness. The results are new and compliment the existing ones.

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28

Bucur, Claudia, and Fausto Ferrari. "An extension problem for the fractional derivative defined by Marchaud." Fractional Calculus and Applied Analysis 19, no. 4 (January 1, 2016). http://dx.doi.org/10.1515/fca-2016-0047.

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AbstractWe prove that the (nonlocal) Marchaud fractional derivative in ℝ can be obtained from a parabolic extension problem with an extra (positive) variable as the operator that maps the heat conduction equation to the Neumann condition. Some properties of the fractional derivative are deduced from those of the local operator. In particular, we prove a Harnack inequality for Marchaud-stationary functions.

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29

Mamatov, T., R. Sabirova, and D. Barakaev. "MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HÖLDER SPACES DEFINED BY USUAL HÖLDER CONDITION." Chronos Journal, no. 11(38) (2019). http://dx.doi.org/10.31618/2658-7556-2019-38-11-1.

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We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class defined by usual Hölder condition

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30

Rafeiro, Humberto, and Makhmadiyor Yakhshiboev. "The Chen-Marchaud fractional integro-differentiation in the variable exponent Lebesgue spaces." Fractional Calculus and Applied Analysis 14, no. 3 (January 1, 2011). http://dx.doi.org/10.2478/s13540-011-0022-8.

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AbstractAfter recalling some definitions regarding the Chen fractional integro-differentiation and discussing the pro et contra of various ways of truncation related to Chen fractional differentiation, we show that, within the framework of weighted Lebesgue spaces with variable exponent, the Chen-Marchaud fractional derivative is the left inverse operator for the Chen fractional integral operator.

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31

Yao, Kui, Haotian Chen, W. L. Peng, Zekun Wang, Jia Yao, and Yipeng Wu. "A new method on Box dimension of Weyl-Marchaud fractional derivative of Weierstrass function." Chaos, Solitons & Fractals, October 2020, 110317. http://dx.doi.org/10.1016/j.chaos.2020.110317.

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32

Priyanka, T. M. C., A. Agathiyan, and A. Gowrisankar. "Weyl–Marchaud fractional derivative of a vector valued fractal interpolation function with function contractivity factors." Journal of Analysis, July 29, 2022. http://dx.doi.org/10.1007/s41478-022-00474-2.

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33

Priyanka, T. M. C., and A. Gowrisankar. "Analysis on Weyl-Marchaud Fractional Derivative for Types of Fractal Interpolation Function with Fractal Dimension." Fractals, July 21, 2021. http://dx.doi.org/10.1142/s0218348x21502157.

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34

Aspenberg, Magnus, Viviane Baladi, Juho Leppänen, and Tomas Persson. "On the fractional susceptibility function of piecewise expanding maps." Discrete & Continuous Dynamical Systems, 2021, 0. http://dx.doi.org/10.3934/dcds.2021133.

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Abstract:

<p style='text-indent:20px;'>We associate to a perturbation <inline-formula><tex-math id="M1">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> of a (stably mixing) piecewise expanding unimodal map <inline-formula><tex-math id="M2">\begin{document}$ f_0 $\end{document}</tex-math></inline-formula> a two-variable fractional susceptibility function <inline-formula><tex-math id="M3">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula>, depending also on a bounded observable <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. For fixed <inline-formula><tex-math id="M5">\begin{document}$ \eta \in (0,1) $\end{document}</tex-math></inline-formula>, we show that the function <inline-formula><tex-math id="M6">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> is holomorphic in a disc <inline-formula><tex-math id="M7">\begin{document}$ D_\eta\subset \mathbb{C} $\end{document}</tex-math></inline-formula> centered at zero of radius <inline-formula><tex-math id="M8">\begin{document}$ >1 $\end{document}</tex-math></inline-formula>, and that <inline-formula><tex-math id="M9">\begin{document}$ \Psi_\phi(\eta, 1) $\end{document}</tex-math></inline-formula> is the Marchaud fractional derivative of order <inline-formula><tex-math id="M10">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> of the function <inline-formula><tex-math id="M11">\begin{document}$ t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t $\end{document}</tex-math></inline-formula>, at <inline-formula><tex-math id="M12">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M13">\begin{document}$ \mu_t $\end{document}</tex-math></inline-formula> is the unique absolutely continuous invariant probability measure of <inline-formula><tex-math id="M14">\begin{document}$ f_t $\end{document}</tex-math></inline-formula>. In addition, we show that <inline-formula><tex-math id="M15">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> admits a holomorphic extension to the domain <inline-formula><tex-math id="M16">\begin{document}$ \{\, (\eta, z) \in \mathbb{C}^2\mid 0<\Re \eta <1, \, z \in D_\eta \,\} $\end{document}</tex-math></inline-formula>. Finally, if the perturbation <inline-formula><tex-math id="M17">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> is horizontal, we prove that <inline-formula><tex-math id="M18">\begin{document}$ \lim_{\eta \in (0,1), \eta \to 1}\Psi_\phi(\eta, 1) = \partial_t \mathcal{R}_\phi(t)|_{t = 0} $\end{document}</tex-math></inline-formula>.</p>

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