The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions

We present the possibilities provided by the MathPartner service of calculating definite and indefinite integrals. MathPartner contains software implementation of the Risch algorithm and provides users with the ability to compute antiderivatives for elementary functions. Certain integrals, including improper integrals, can be calculated using numerical algorithms. In this case, every user has the ability to indicate the required accuracy with which he needs to know the numerical value of the integral. We highlight special functions allowing us to calculate complete elliptic integrals. These include functions for calculating the arithmetic-geometric mean and the geometric-harmonic mean, which allow us to calculate the complete elliptic integrals of the first kind. The set also includes the modified arithmetic-geometric mean, proposed by Semjon Adlaj, which allows us to calculate the complete elliptic integrals of the second kind as well as the circumference of an ellipse. The Lagutinski algorithm is of particular interest. For given differentiation in the field of bivariate rational functions, one can decide whether there exists a rational integral. The algorithm is based on calculating the Lagutinski determinant. This year we are celebrating 150th anniversary of Mikhail Lagutinski.

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The GMD Method for Inductance Calculation Applied to Conductors with Skin Effect

Advanced Electromagnetics ◽

10.7716/aem.v6i2.464 ◽

2017 ◽

Vol 6 (2) ◽

pp. 77

Author(s):

H. A. Aebischer ◽

B. Aebischer

Keyword(s):

Cross Section ◽

Skin Effect ◽

Unit Length ◽

Geometric Mean ◽

Analytical Approximation ◽

Elementary Functions ◽

Exact Relation ◽

Measurement Results ◽

Definition Of ◽

Analytical Expressions

The GMD method (geometric mean distance) to calculate inductance offers undoubted advantages over other methods. But so far it seemed to be limited to the case where the current is uniformly distributed over the cross section of the conductor, i.e. to DC (direct current). In this paper, the definition of the GMD is extended to include cases of nonuniform distribution observed at higher frequencies as the result of skin effect. An exact relation between the GMD and the internal inductance per unit length for infinitely long conductors of circularly symmetric cross section is derived. It enables much simpler derivations of Maxwell’s analytical expressions for the GMD of circular and annular disks than were known before. Its salient application, however, is the derivation of exact expressions for the GMD of infinitely long round wires and tubular conductors with skin effect. These expressions are then used to verify the consistency of the extended definition of the GMD. Further, approximate formulae for the GMD of round wires with skin effect based on elementary functions are discussed. Total inductances calculated with the help of the derived formulae for the GMD with and without skin effect are compared to measurement results from the literature. For conductors of square cross section, an analytical approximation for the GMD with skin effect based on elementary functions is presented. It is shown that it allows to calculate the total inductance of such conductors for frequencies from DC up to 25 GHz to a precision of better than 1 %.

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Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions

Journal of Applied Mathematics ◽

10.1155/2012/286290 ◽

2012 ◽

Vol 2012 ◽

pp. 1-16 ◽

Cited By ~ 9

Author(s):

A. Beléndez ◽

M. L. Alvarez ◽

J. Francés ◽

S. Bleda ◽

T. Beléndez ◽

...

Keyword(s):

Elliptic Integral ◽

Duffing Oscillator ◽

Geometric Mean ◽

Harmonic Balance Method ◽

Approximate Solutions ◽

Jacobi Elliptic Function ◽

Complete Elliptic Integral ◽

Elementary Functions ◽

Restoring Force ◽

Closed Form Solutions

Accurate approximate closed-form solutions for the cubic-quintic Duffing oscillator are obtained in terms of elementary functions. To do this, we use the previous results obtained using a cubication method in which the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation. Explicit approximate solutions are then expressed as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn. Then we obtain other approximate expressions for these solutions, which are expressed in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean is used and the rational harmonic balance method is applied to obtain the periodic solution of the original nonlinear oscillator.

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Radiation synovectomy of the knee joint

Nuklearmedizin ◽

10.1055/s-0038-1623935 ◽

2006 ◽

Vol 45 (01) ◽

pp. 57-61

Author(s):

M. Puille ◽

D. Steiner ◽

R. Bauer ◽

R. Klett

Keyword(s):

Knee Joint ◽

Lymph Nodes ◽

Geometric Mean ◽

Radiation Synovectomy ◽

Real Activity ◽

Anterior View ◽

Inguinal Lymph Nodes ◽

The Real ◽

Therapeutic Modalities ◽

Real Value

Summary Aim: Multiple procedures for the quantification of activity leakage in radiation synovectomy of the knee joint have been described in the literature. We compared these procedures considering the real conditions of dispersion and absorption using a corpse phantom. Methods: We simulated different distributions of the activity in the knee joint and a different extra-articular spread into the inguinal lymph nodes. The activity was measured with a gammacamera. Activity leakage was calculated by measuring the retention in the knee joint only using an anterior view, using the geometric mean of anterior and posterior views, or using the sum of anterior and posterior views. The same procedures were used to quantify the activity leakage by measuring the activity spread into the inguinal lymph nodes. In addition, the influence of scattered rays was evaluated. Results: For several procedures we found an excellent association with the real activity leakage, shown by an r² between 0.97 and 0.98. When the real value of the leakage is needed, e. g. in dosimetric studies, simultaneously measuring of knee activity and activity in the inguinal lymph nodes in anterior and posterior views and calculation of the geometric mean with exclusion of the scatter rays was found to be the procedure of choice. Conclusion: When measuring of activity leakage is used for dosimetric calculations, the above-described procedure should be used. When the real value of the leakage is not necessary, e. g. for comparing different therapeutic modalities, several of the procedures can be considered as being equivalent.

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DMSA-scintigraphy in paediatrics: Is the evaluation of the geometric mean necessary for the calculation of the differential renal function?

Nuklearmedizin ◽

10.1055/s-0038-1625921 ◽

2001 ◽

Vol 40 (04) ◽

pp. 107-110 ◽

Cited By ~ 1

Author(s):

B. Roßmüller ◽

S. Alalp ◽

S. Fischer ◽

S. Dresel ◽

K. Hahn ◽

...

Keyword(s):

Renal Function ◽

Geometric Mean ◽

Region Of Interest ◽

Posterior View ◽

Renal Abnormality ◽

Anterior View ◽

Differential Renal Function ◽

The Mean ◽

The Right ◽

To Receive

SummaryFor assessment of differential renal function (PF) by means of static renal scintigraphy with Tc-99m-dimer-captosuccinic acid (DMSA) the calculation of the geometric mean of counts from the anterior and posterior view is recommended. Aim of this retrospective study was to find out, if the anterior view is necessary to receive an accurate differential renal function by calculating the geometric mean compared to calculating PF using the counts of the posterior view only. Methods: 164 DMSA-scans of 151 children (86 f, 65 m) aged 16 d to 16 a (4.7 ± 3.9 a) were reviewed. The scans were performed using a dual head gamma camera (Picker Prism 2000 XP, low energy ultra high resolution collimator, matrix 256 x 256,300 kcts/view, Zoom: 1.6-2.0). Background corrected values from both kidneys anterior and posterior were obtained. Using region of interest technique PF was calculated using the counts of the dorsal view and compared with the calculated geometric mean [SQR(Ctsdors x Ctsventr]. Results: The differential function of the right kidney was significantly less when compared to the calculation of the geometric mean (p<0.01). The mean difference between the PFgeom and the PFdors was 1.5 ± 1.4%. A difference > 5% (5.0-9.5%) was obtained in only 6/164 scans (3.7%). Three of 6 patients presented with an underestimated PFdors due to dystopic kidneys on the left side in 2 patients and on the right side in one patient. The other 3 patients with a difference >5% did not show any renal abnormality. Conclusion: The calculation of the PF from the posterior view only will give an underestimated value of the right kidney compared to the calculation of the geometric mean. This effect is not relevant for the calculation of the differntial renal function in orthotopic kidneys, so that in these cases the anterior view is not necesssary. However, geometric mean calculation to obtain reliable values for differential renal function should be applied in cases with an obvious anatomical abnormality.

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On Statistics of Log-Ratio of Arithmetic Mean to Geometric Mean for Nakagami-m Fading Power

IEICE Transactions on Communications ◽

10.1587/transcom.e95.b.647 ◽

2012 ◽

Vol E95-B (2) ◽

pp. 647-650

Author(s):

Ning WANG ◽

Julian CHENG ◽

Chintha TELLAMBURA

Keyword(s):

Geometric Mean ◽

Arithmetic Mean ◽

Log Ratio

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