Their coefficients are expressed in a power series of the orbital eccentricity e and extended up to eighth-order terms of the orbital eccentricity. Let the mean anomaly be M. M can be expressed by E as follows:.

### Mathematical Analysis, Modelling, and Applications

Differentiating the both sides of Eq. To expand Eq. It is easy to expand Eq. One can imagine that these procedures are too complicated to be realized by hand, but will become much easier and be significantly simplified by means of Mathematica. Omitting the detailed procedure in Mathematica, one arrives at.

Integrating at the both sides of Eq. From Eq. In order to validate the exactness of the derived series expansions, one has examined their accuracies when the orbital eccentricity e is respectively equal to 0. Substituting M 0 into Eq. Substituting E 0 into Eq. Due to limited space, these errors when e is equal to 0. Other numerical examples indicate that when the orbital eccentricity e is respectively equal to 0.

Some typical mathematical problems in geodesy are solved by means of computer algebra analysis method and computer algebra system Mathematica. The main contents and research results presented in this chapter are as follows:. The forward and inverse expansions of the meridian arc often used in geometric geodesy are derived. Their coefficients are expressed in a power series of the first eccentricity of the reference ellipsoid and extended up to its tenth-order terms.

The singularity existing in the integral expressions of height anomaly, deflections of the vertical and gravity gradient is eliminated using the nonsingular integration transformations, and the nonsingular expressions are systematically derived. The series expansions of direct transformations between three anomalies in satellite geodesy are derived using the power series method.

We would like to express our great appreciation to the editor and reviewers. Thanks very much for the kind work and consideration of Ms. Kristina Jurdana on the publication of this chapter. Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3. Help us write another book on this subject and reach those readers. Login to your personal dashboard for more detailed statistics on your publications.

Edited by Rifaat Abdalla. Edited by Pasquale Imperatore. We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals.

Downloaded: Keywords geodesy computer algebra mathematical analysis meridian arc singular integration mean anomaly. Introduction Geodesy is the science of accurately measuring and understanding three fundamental properties of the Earth: its geometric shape, its orientation in space, and its gravity field, as well as the changes of these properties with time. The forward and inverse expansions of the meridian arc in geometric geodesy The forward and inverse problem of the meridian arc is one of the fundamental problems in geometric geodesy, which seems to be a solved one.

The inverse expansion of the meridian arc using the Hermite interpolation method Differentiation to the both sides of Eq. Table 1. Errors of the inverse expansions of meridian arc. The singular integration in physical geodesy Singular integrals associated with the reciprocal distance usually exist in the computations of physical geodesy and geophysics.

The main contents and research results presented in this chapter are as follows:. The forward and inverse expansions of the meridian arc often used in geometric geodesy are derived.

Their coefficients are expressed in a power series of the first eccentricity of the reference ellipsoid and extended up to its tenth-order terms. The singularity existing in the integral expressions of height anomaly, deflections of the vertical and gravity gradient is eliminated using the nonsingular integration transformations, and the nonsingular expressions are systematically derived.

The series expansions of direct transformations between three anomalies in satellite geodesy are derived using the power series method. We would like to express our great appreciation to the editor and reviewers.

## Problems in Mathematical Analysis III: Integration

Thanks very much for the kind work and consideration of Ms. Kristina Jurdana on the publication of this chapter. Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3. Help us write another book on this subject and reach those readers. Login to your personal dashboard for more detailed statistics on your publications. Edited by Rifaat Abdalla. Edited by Pasquale Imperatore. We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals.

Downloaded: Keywords geodesy computer algebra mathematical analysis meridian arc singular integration mean anomaly. Introduction Geodesy is the science of accurately measuring and understanding three fundamental properties of the Earth: its geometric shape, its orientation in space, and its gravity field, as well as the changes of these properties with time. The forward and inverse expansions of the meridian arc in geometric geodesy The forward and inverse problem of the meridian arc is one of the fundamental problems in geometric geodesy, which seems to be a solved one.

The inverse expansion of the meridian arc using the Hermite interpolation method Differentiation to the both sides of Eq. Table 1. Errors of the inverse expansions of meridian arc. The singular integration in physical geodesy Singular integrals associated with the reciprocal distance usually exist in the computations of physical geodesy and geophysics.

The series expansions of direct transformations between three anomalies in satellite geodesy The determination of satellite orbit is one of the fundamental problems in satellite geodesy. The series expansions of the direct transformation between eccentric and mean anomalies Let the mean anomaly be M.

### 1st Edition

The accuracy of the derived series expansions In order to validate the exactness of the derived series expansions, one has examined their accuracies when the orbital eccentricity e is respectively equal to 0. Table 2. Errors of the derived s eries expansions. Conclusions Some typical mathematical problems in geodesy are solved by means of computer algebra analysis method and computer algebra system Mathematica.

The main contents and research results presented in this chapter are as follows: The forward and inverse expansions of the meridian arc often used in geometric geodesy are derived. Conflict of interest There are no conflicts of interest. Thanks We would like to express our great appreciation to the editor and reviewers.

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