Research articles authored by Henrik Stenlund

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Henrik Stenlund on ResearchGate

Henrik Stenlund,** **"On the Elementary Trigonometric Polynomials and their Generating Functions", Quest Journals Journal of Research in Applied Mathematics Volume 8 ~ Issue 11 (2022) pp: 08-15 **(local copy)**

Visilab Report #2022-11

The trigonometric polynomials are simple yet have lots of interest in theories of orthogonal polynomials and in numerical analysis. The elementary trigonometric polynomials are broken series of sin(x) and cos(x). Some of their important properties have been left unsolved. Their integral representations are derived and the polynomials can be generated with them. Their differential equations are solved and the generating functions are created with the help of their recursion relations. A correspondence was observed between the exponential and trigonometric polynomials in the form of a pair of equations.

Henrik Stenlund, "A Note on the Exponential Polynomial and Its Generating Function", Quest Journals Journal of Research in Applied Mathematics Volume 8 ~ Issue 7 (2022) pp: 28-30 (local copy)

Visilab Report #2022-07

The basic exponential polynomial is simple but has no closed-form expression other than the polynomial itself. This polynomial is mostly studied in group theories and with prime numbers. However, some of its basic properties seem to have been ignored. In this paper are shown an integral representation for it with which one can generate the polynomial. It is based on both a recursion relation and a differential equation. By using them various generating functions are solved.

Henrik Stenlund,** "**On Transforming Functions of a Certain Dot-Product Gradient Operator**"**, arXiv:2107.12155v1 [math.GM] 21 July 2021 **(local copy)**

Visilab Report #2021-07

In this paper it is shown that a function of the constant dot product of the gradient operator acting on an arbitrary function can be transformed to a double three-dimensional integral. The inner one of them is a Fourier transform of the operator function. The result converted to one-dimensional problems is also useful in transforming complex differential expressions.

Henrik Stenlund,** "****On Transforming the Laplace Operator****"**, IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 17, Issue 3 Ser. I (May – June 2021), PP 59-64 www.iosrjournals.org 8 May 2021 ** (local copy)**

Visilab Report #2021-05

In this communication it is shown that a function of the Laplace operator acting on an arbitrary function can be transformed to a three-dimensional integral. The cases of the exponential function and of an arbitrary function expressible as a power series, are treated. Two special cases of radial functions are presented based on elementary observations made here.

Henrik Stenlund,** "****Connection between the Inversion Formula and the Functional Power Series"**, IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 6 Ser. II (Nov. – Dec. 2020), PP 22-25 www.iosrjournals.org 12 Mar 2020 ** (local copy)**

Visilab Report #2020-11

This study derives the connection between the functional power series and the inversion formula, both given in series form. The link is established by differentiation of the inversion formula showing an expression of a functional power series for the reciprocal of the derivative in terms of the function itself.

Henrik Stenlund,** "****On the Basics of Linear Diffusion"**, IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 2 Ser. III (Mar. – Apr. 2020), PP 44-49 www.iosrjournals.org 12 Mar 2020 ** (local copy)**

Visilab Report #2020-04

This study handles the three-dimensional linear diffusion in a new way. A general solution is given without particular initial conditions. In addition, solutions are obtained for a source-sink as a constant in time but spatially varying in three dimensions and having an arbitrary time dependence. An auxiliary function for diffusion is given having an interesting relationship with the concentration. It appears that both the time derivative and the Laplacian of the concentration obey the diffusion equation. An integro-differential equation for diffusion is presented.

Henrik Stenlund,** "****On the Basics of the Nonlinear Diffusion"**, arXiv:2003.06282v1 [math.GM] 12 Mar 2020 ** (local copy)**

Visilab Report #2020-03

This study handles spatial three-dimensional solution of the nonlinear diffusion equation without particular initial conditions. The functional behavior of the equation and the concentration have been studied in new ways. An auxiliary function for diffusion is given having an interesting relationship with the concentration. A set of new integro-differential equations is given for diffusion

Henrik Stenlund,** "On Relations Between the Stirling Numbers of First and Second Kind"**, arXiv:1903.11947v1 [math.GM] 25 Mar 2019 ** (local copy)**

Visilab Report #2019-04

Four new relations have been found between the Stirling numbers of first and second kind. They are derived directly from recently published relations.

Henrik Stenlund,** "On Some Relations Between the Stirling Numbers of First and Second Kind"**, International Journal of Mathematics and Computer Research ISSN: 2320-7167 Volume 07 Issue 03 March-2019, Page no. - 1948-1950 Index Copernicus ICV: 57.55 DOI: 10.31142/ijmcr/v7i03.01**(local copy)**

Visilab Report #2019-03

The powers of the ordinary differential operator can be expanded in terms of the Cauchy-Euler differential operator and for the opposite case. The expansions involve the Stirling numbers of first and second kind as is well known. Two relations between the Stirling numbers of first and second kind will find their proof in this work, generated by the two expansions. A third relation is obtained by algebraic manipulation from the two known recursion relations.

Henrik Stenlund,** "On Studying the Phase Behavior of the Riemann Zeta Functions Along the Critical Line", arXiv:1806.011148v1 [math.GM]** 1 Jun 2018 ** (local copy)**

Visilab Report #2018-06

The critical line of the Riemann zeta function is studied from a new viewpoint. It is found that the ratio between the zeta function at any zero and the corresponding one at a conjugate point has a certain phase and its absolute value is unity. This fact is valid along the whole critical line and only there. The common functional equation is used with the aid of the function ratio between any zero and its negative side pair, a complex conjugate. As a result, an equation is obtained for solving the phase along the critical line.

Henrik Stenlund,**"Theory Classical Gaussian Observer", Global Journal of of Science Frontier Research: F mathematics and Decision Sciences, Volume 17,Issue 7 version 1.0, Year 2017, ** ** (local copy)**

Visilab Report #2017-11

- This paper treats the concept of the Gaussian probability distribution both for the target and observer. The resulting observations become Gaussian distributions as well. The time coordinate gets an equal setting as any physical quantity. This treatment is purely classical with no essential reference to quantum mechanics nor to theory of relativity.

Henrik Stenlund,**"On Vector Functions With A Parameter", Journal of Research in Applied Mathematics, Volume 3, no. 6, 2017, ** ** (local copy)**

Visilab Report #2017-07

- A set of equations for removing and adding of a parameter were found in a scalar type vector function. By using the Cauchy-Euler differential operator in an exponential form, equations for calculation of the partial derivative with respect to the parameter were developed. Extensions to multiple vector and higher rank functions were made.

Henrik Stenlund,** "On Transforming the Generalized Exponential Series", arXiv:1701.00515v1 [math.GM]** 27 Dec 2016 ** (local copy)**

Visilab Report #2016-12

- We transformed the generalized exponential power series to another functional form suitable for further analysis. By applying the Cauchy-Euler differential operator in the form of an exponential operator, the series became a sum of exponential differential operators acting on a simple exponential (e(-x)). In the process we found new relations for the operator and a new polynomial with some interesting properties. Another form of the exponential power series became a nested sum of the new polynomial, thus isolating the main variable to a different functional dependence. We studied shortly the asymptotic behavior by using the dominant terms of the transformed series. New series expressions were created for common functions, like the trigonometric and exponential functions, in terms of the polynomial.

Henrik Stenlund,** "On Infinite Identities Generating Solutions for Series", arXiv:1611.01375v2 [math.GM]** 30 Oct 2016 ** (local copy)**

Visilab Report #2016-11

- In this paper we present a new identity and some of its variants which can be used for finding solutions while solving fractional infinite and finite series. We introduce another simple identity which is capable of generating solutions for some finite series. We demonstrate a method for generation of variants of the identities based on the findings. The identities are useful for solving various infinite products..

Henrik Stenlund,**"Methods for the Summation of Infinite Series", International Journal of Mathematics and Computer Science, Volume 11, no. 2, 2016, ** ** (local copy)**

Visilab Report #2016-08

- This paper is about methods for expressing infinite series in closed form by using Laplace transforms and their inverses where resulting integrals are to be evaluated instead. If needed, parametrizing the series and applying an additional Laplace transform offer more options for the summation of infinite series

Henrik Stenlund,** "On Methods for Transforming and Solving Finite Series", arXiv:1602.04080v1 [math.GM]** 28 Jan 2016 ** (local copy)**

Visilab Report #2016-01

- In this work we present new methods for transforming and solving finite series by using the Laplace transform. In addition we introduce both an alternative method based on the Fourier transform and a simplified approach. The latter allows a quick solution in some cases.

Henrik Stenlund, ** "On Cauchy-Euler Operator", 20th November 2015**

**DOI:** 10.13140/RG.2.1.4197.1285

Visilab Report #2015-11

- We present some results obtained for the Cauchy-Euler differential operator, especially for the exponential function of it.

Henrik Stenlund, ** "A Closed-Form Solution to the Arbitrary Order Cauchy Problem with Propagators " , arXiv:1411.6890v1[math.GM] , 24th November 2014 ** ** (local copy)**

Visilab Report #2014-12

- The general abstract arbitrary order (N) Cauchy problem was solved in a closed form as a sum of exponential propagator functions. The infinite sparse exponential series was solved with the aid of a homogeneous differential equation. It generated a linear combination of exponential functions. The Cauchy problem solution was formed with N linear combinations of N exponential propagators.

Henrik Stenlund,**"On Solving the Cauchy Problem with Propagators", arXiv:1411.1402v1[math.AP], 5th November 2014 ** ** (local copy)**

Visilab Report #2014-11

- The abstract first order Cauchy problem is solved in terms of Taylor’s series leading to a series of operators which is a propagator. It is found that higher order Cauchy problems can be solved in the same way. Since derivatives of order lower than the Cauchy problem are built-in to the problem, it suffices to solve the higher order derivatives in terms of the lower order ones, in a simple way.

Henrik Stenlund,**"On a Method for Solving Infinite Series", arXiv:1405.7633v2[math.GM], 6th May 2014 ** ** (local copy)**

Visilab Report #2014-05

- This paper is about a method for solving infinite series in closed form by using inverse and forward Laplace transforms. The resulting integral is to be solved instead. The method is extended by parametrizing the series. A further Laplace transform with respect to it will offer more options for solving a series.

Henrik Stenlund, **"A Note on Laplace Transforms of Some Particular Function Types", arXiv:1402.2876v1[math.GM],**** 9th Feb 2014 ** ** (local copy)**

Visilab Report #2014-02

- This article handles in a short manner a few Laplace transform pairs and some extensions to the basic equations are developed. They can be applied to a wide variety of functions in order to find the Laplace transform for its inverse when there is a specific type of an implicit function involved .

Henrik Stenlund, **"On Solving Some Trigonometric Series", arXiv:1308.2626v1[math.GM],**** 5th Aug 2013 ** ** (local copy)**

Visilab Report #2013-08

- This communication shows the track for finding a solution for a sin(kx)/k**2 series and a fresh representation for the Euler's Gamma function in terms of Riemann's Zeta function. We have found a new series expression for the logarithm as a side effect. The new series are useful both for analysis, approximations and asymptotic studies.

Henrik Stenlund, ** "Functional Power Series", arXiv:1204.5992[math.GM], 24th April 2012 ** ** (local copy)**

Visilab Report #2012-04

- This work introduces a new functional series for expanding an analytic function in terms of an arbitrary analytic function. It is generally applicable and straightforward to use. It is also suitable for approximating the behavior of a function with a few terms. A new expression is presented for the composite function's n'th derivative. The inverse-composite method is handled in this work also.

Henrik Stenlund, ** "Recursion Relations and Functional Equations for the Riemann Zeta Function", arXiv:1107.3479v3[math.GM], 18th Jul 2011 ** ** (local copy)**

Visilab Report #2011-07

- New recursion relations for the Riemann zeta function are introduced. Their derivation started from the standard functional equation. The new functional equations have both real and imaginary increment versions and can be applied over the whole complex plane. We have developed various versions of the recursion relations eliminating each of the coefficient functions, leaving plain zeta functions.

Henrik Stenlund, ** "The Inversion Formula Applied to Some Examples , 1st Jan 2011 **

Visilab Report #2011-01

- This work discusses a few applications of the recent inversion formula to various functions. The examples range from elementary to more involved. The purpose is to indicate the way of using the inversion formula in general. The target audience is among physicists, chemists, biologists and in other sciences with less inclination to formal mathematics. This article is subject to future modifications.

Henrik Stenlund, ** "Inversion Formula", arXiv:1008.0183v3[math.GM], 27th Jul 2010 ** ** (local copy)**

Visilab Report #2010-07

- This work introduces a new inversion formula for analytical functions. It is simple, generally applicable and straightforward to use both in hand calculations and for symbolic machine processing. It is easier to apply than the traditional Lagrange-Bürmann formula since no taking limits is required. This formula is important for inverting functions in physical and mathematical problems.

Henrik Stenlund, ** "Technical Report: "Three Methods of Solution of Concentration Dependent Diffusion Coefficient", 6-18-2004**

**DOI:** 10.13140/RG.2.1.5147.4006

- This study offers three solutions of the diffusion coefficient's dependence on concentration in general cases without any limitations by boundary conditions. They are all suitable for numerical analysis when the experimental concentration data and time series are available producing dependence functions. As they are also of general nature, the expressions can be used for further investigations and modeling and fitting. Two of the methods offer three-dimensional approaches to this problem and may prove useful when combined with present-day laser scanning volumetric sensors, atomic probe microscopes and high performance computers. This is particularly true in geometries more complex than the regular one consisting of two semi-infinite slabs.

Henrik Stenlund, **"Report on Optimization of an Optical System", 01-15-2001**

- A detailed study for optimizing an optical multichannel detector system for a colour measuring system. The distance dependence was the most important parameter. Monte Carlo method was used in ray tracing.

Henrik Stenlund, ** "Technical Report: A Research on Mutual Interaction of Two Electron Beams via Space Charge", 12-13-1999**

- A study of the mutual interaction of two electron beams in a vacuum CRT. Actual application in the effects found at a CRT factory.

Henrik Stenlund, "**Technical Report: Thin Cassette Oven for GC with a Silica Column**", 03-12-1990

- A summary report of the development project for a high temperature oven. The target system was a thin small cassette for a narrow Silica gas chromatographic (GC) column. The temperature range is from +50 to +400 C with a very high control accuracy and very small temperature gradients inside the cassette. Theoretical work around thermal conductivity and temperature distributions of the cassette formed the basis. According to those results a few prototypes were built and tested. Verification of proper operation was made with true GC runs.

Henrik Stenlund, "**Technical Report: Measuring Paper Formation with Beta Radiation**", KCL P8622, 06-19-1986

- A theoretical report on measuring paper formation with beta radiation. Radioactive intensity calculations for required signal level are established with formulas.

Henrik Stenlund, "**Technical Report: Measuring the Hydrodynamic Pressure in a Wet Pressing Process**",KCL P8630 06-03-1986

- A theoretical report on measuring the hydrodynamic pressure of water and fiber mixture in a wet press for paper production. Part 3.

Henrik Stenlund, "**Technical Report: Design and Implementation of Time-of-Flight Mass Spectrometer**", TEKES 105/425/84;119/1985, 3-30-1986 (in finnish)

- Final report of a project for designing two differing TOFMS of optimized reflection type. All technical details of electronics, mechanics and software are shown.

The purpose of the TOFMS project was to develop a prototype being feasible for manufacturing as a product, having sensible spectrometric features. A compact structure for the instrument was designed by making it in axial form. As a result we have two prototypes both having mass resolution much better than R > 1500 and the useful mass range over 1200 amu. The original specification was to exceed R = 400. The system is able to deliver 5000 spectra / s on the screen of an oscilloscope. The free length of flight was about 1000 mm and the ion optical mechanical parts required some 800 mm in the vacuum chamber.

Henrik Stenlund, Karl Holmström, "**Technical Report: Using Microwaves for Measuring Blade Play in a Fiber Grinding Machine**", KCL P8636, 02-09-1986

- A preliminary report on feasibility of using microwaves for measuring blade play in a grinder. Continuous measurement would allow replacement of the blade in time and save in fiber quality, energy and productivity.

Henrik Stenlund, "**Technical Report: Design of the Vacuum System for the Wet Press Simulator Instrument**", KCL P8630 01-20-1986

- A vacuum pumping station was designed for measuring the hydrodynamic pressure of water and fiber mixture in a wet press for paper production. Pumping equations were solved and recommendations were given for proper building of the system.

Henrik Stenlund, "**Technical Report: Measuring the Hydrodynamic Pressure in a Wet Pressing Process**", KCL S8630 01-10-1986

- A theoretical report on measuring the hydrodynamic pressure of water and fiber mixture in a wet press for paper production. Part 2.

Henrik Stenlund, "**Technical Report: Measuring the Hydrodynamic Pressure in a Wet Pressing Process**", KCL S8545 12-04-1985

- A theoretical report on measuring the hydrodynamic pressure of water and fiber mixture in a wet press for paper production. Part 1.

Henrik Stenlund, Department of Chemistry, University of Helsinki, "**Time-of-Flight Mass Spectrometry**", Finnish Physical Society Meeting, Oulu, 2-8-1985

- A prototype of a reflection type TOFMS is presented.

Henrik Stenlund, "**Technical Report: Design of an Ultrasound Sensor Head for Sinuscan Sinuitis Diagnosing Device**", Orion Pharma Oy, Orion Analytica, 05-08-1984

- a feasibility study over an ultrasound transducer for the Sinuscan brand analyzer. An US sensor was studied both in theory and practice. As a result, manufacturing guidance was established for the product.

Henrik Stenlund, "**Theory of Ionization**", a lecture presented at Orion Pharma Oy, Orion Analytica, 02-19-1981

- a lecture over various ionization methods for gases was given. Also some ion mass separation techniques were explained.

Henrik Stenlund, ** "Design of a High Intensity Electron Gun and Its Focusing System", Res. rep. 2/1980, Helsinki University of Technology, Laboratory of Physics **

- Electron optical system design with magnetic lenses for an X-ray system. The spherical electron gun design is also described in detail.

Henrik Stenlund, **"Quantum Theory of Interstitial Atomic Diffusion of Light Impurities in Semiconductors", University of Helsinki, 1979, a Ph.Lic thesis**

A quantum theory of atomic diffusion in solids is presented and applied to a particular problem of interstitial diffusion, Li in Ge and Si. The theory is based on transport in narrow bands using the relaxation time approximation. The bands are evaluated by a one-dimensional method and the interaction potential between the impurity atom and lattice atoms is examined through numerical calculations, where the lattice potential difference between the equilibrium site and saddle point, the potential barrier height, is fitted with experimental values.

Henrik Stenlund, ** "Quantum Theory of Interstitial Diffusion", University of Helsinki, 1979, a private study**

A quantum theory of atomic diffusion in solids is presented and applied to a particular problem of interstitial diffusion, Li in Ge and Si. The theory is based on transport in allowed narrow bands using the relaxation time approximation. The bands were calculated by using a proper potential model through numerical methods. The relaxation time was taken from the theory of Kagan and Klinger [18] and the diffusion coefficients were calculated. For comparison, the result of the theory of Kagan and Klinger was put also in numerical form. All coefficients show relatively good agreement with experimental values (see table 3.).The isotopic effect was also studied and Do and Eo were found to have dependences of (m)**-1/2 and (m)**-1 + B, respectively.