scholarly journals On the generalized order and generalized type of entire monogenic functions

2013 ◽  
Vol 46 (4) ◽  
Author(s):  
G. S. Srivastava ◽  
Susheel Kumar
Keyword(s):  
Lower Order ◽  
Monogenic Functions ◽  
Generalized Type ◽  
Taylor’S Series ◽  
Generalized Order

AbstractIn the present paper we study the generalized growth of entire monogenic functions. The generalized order, generalized lower order and generalized type of entire monogenic functions have been obtained in terms of its Taylor’s series coefficients.

scholarly journals Generalized Growth of Special Monogenic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Susheel Kumar
Keyword(s):  
Lower Order ◽  
Monogenic Functions ◽  
Lower Type ◽  
Generalized Type ◽  
Taylor’S Series ◽  
Generalized Order

We study the generalized growth of special monogenic functions. The characterizations of generalized order, generalized lower order, generalized type, and generalized lower type of special monogenic functions have been obtained in terms of their Taylor’s series coefficients.

Generalized slow growth of special monogenic functions

2016 ◽  
Vol 22 (1) ◽  
Author(s):  
Susheel Kumar
Keyword(s):  
Taylor Series ◽  
Lower Order ◽  
Slow Growth ◽  
Monogenic Functions ◽  
Lower Type ◽  
Generalized Type ◽  
Generalized Order

AbstractIn the present paper we study the generalized slow growth of special monogenic functions. The characterizations of generalized order, generalized lower order, generalized type and generalized lower type of special monogenic functions have been obtained in terms of their Taylor series coefficients.

scholarly journals Coefficients Characterization of Entire Harmonic Functions in Terms of Norm of Gradients at Origin in R^n, n≥ 3

2020 ◽  
Vol 13 (2) ◽  
pp. 258-268
Author(s):  
Devendra Kumar ◽  
Rajeev Kumar Vishnoi
Keyword(s):  
Harmonic Function ◽  
Lower Order ◽  
Spherical Harmonic ◽  
Harmonic Functions ◽  
Spherical Harmonic Expansion ◽  
Generalized Type ◽  
Generalized Order

Coefficient characterizations of generalized order, lower order and generalized type of entire harmonic function having the spherical harmonic expansion throughout a neighborhood of the origin in Rn have been obtained in terms of norm of gradients at origin.

On the Lower Order and Type of Entire Axially Monogenic Functions

2012 ◽  
Vol 63 (3-4) ◽  
pp. 1257-1275 ◽  
Author(s):  
M. A. Abul-Ez ◽  
R. De Almeida
Keyword(s):  
Lower Order ◽  
Monogenic Functions ◽  
Order And Type

On the Lower Order of Entire Special Monogenic Functions

2011 ◽  
Author(s):  
M. A. Abul-Ez ◽  
R. De Almeida ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
...  
Keyword(s):  
Lower Order ◽  
Monogenic Functions

scholarly journals RELATIVE GROWTH OF ENTIRE DIRICHLET SERIES WITH DIFFERENT GENERALIZED ORDERS

2021 ◽  
Vol 9 (2) ◽  
pp. 22-34
Author(s):  
M. Sheremeta ◽  
O. Mulyava
Keyword(s):  
Dirichlet Series ◽  
Lower Order ◽  
Entire Functions ◽  
Entire Dirichlet Series ◽  
Relative Growth ◽  
Alpha Beta ◽  
Generalized Order

For entire functions $F$ and $G$ defined by Dirichlet series with exponents increasing to $+\infty$ formulas are found for the finding the generalized order $\displaystyle \varrho_{\alpha,\beta}[F]_G = \varlimsup\limits_{\sigma\to=\infty} \frac{\alpha(M^{-1}_G(M_F(\sigma)))}{\beta(\sigma)}$ and the generalized lower order $\displaystyle \lambda_{\alpha,\beta}[F]_G=\varliminf\limits_{\sigma\to+\infty} \frac{\alpha(M^{-1}_G(M_F(\sigma)))}{\beta(\sigma)}$ of $F$ with respect to $G$, where $M_F(\sigma)=\sup\{|F(\sigma+it)|:\,t\in{\Bbb R}\}$ and $\alpha$ and $\beta$ are positive increasing to $+\infty$ functions.

scholarly journals GENERALIZED ORDER \((\alpha ,\beta)\) ORIENTED SOME GROWTH PROPERTIES OF COMPOSITE ENTIRE FUNCTIONS

2020 ◽  
Vol 6 (2) ◽  
pp. 25
Author(s):  
Tanmay Biswas ◽  
Chinmay Biswas
Keyword(s):  
Lower Order ◽  
Entire Functions ◽  
Growth Properties ◽  
Left And Right ◽  
Alpha Beta ◽  
Generalized Order

In this paper we establish some results relating to the growths of composition of two entire functions with their corresponding left and right factors on the basis of their generalized order \((\alpha ,\beta )\) and generalized lower order \((\alpha ,\beta )\) where \(\alpha \) and \(\beta \) are continuous non-negative functions on \((-\infty ,+\infty )\).

scholarly journals Generalized order of entire monogenic functions of slow growth

2012 ◽  
Vol 05 (06) ◽  
pp. 418-425 ◽  
Author(s):  
Susheel Kumar ◽  
Kirandeep Bala
Keyword(s):  
Slow Growth ◽  
Monogenic Functions ◽  
Generalized Order

scholarly journals On the relative growth of Dirichlet series with zero abscissa of absolute convergence

2021 ◽  
Vol 55 (1) ◽  
pp. 44-50
Author(s):  
O. M. Mulyava
Keyword(s):  
Dirichlet Series ◽  
Lower Order ◽  
Analytic Functions ◽  
Absolute Convergence ◽  
Relative Growth ◽  
Alpha Beta ◽  
Function Inverse ◽  
Generalized Order

Let $F$ and $G$ be analytic functions given by Dirichlet series with exponents increasing to $+\infty$ and zero abscissa of absolute convergence.The growth of $F$ with respect to $G$ is studied through the generalized order$$\varrho^0_{\alpha,\beta}[F]_G=\varlimsup\limits_{\sigma\uparrow 0}\dfrac{\alpha(1/|M^{-1}_G(M_F(\sigma)|)}{\beta(1/|\sigma|)}$$and the generalized lower order $$\lambda^0_{\alpha,\beta}[F]_G=\varliminf\limits_{\sigma\uparrow 0} \dfrac{\alpha(1/|M^{-1}_G(M_F(\sigma)|)}{\beta(1/|\sigma|)},$$ where $M_F(\sigma)=\sup\{|F(\sigma+it)|:\,t\in{\mathbb R}\},$ $M^{-1}_G(x)$ is the function inverse to $M_G(\sigma)$ and $\alpha$ and $\beta$ are positive increasing to $+\infty$ functions.Formulas are found for the finding these quantities.

Experiments with a Quadrupole Octopole Corrector in a STEM

1977 ◽  
Vol 35 ◽  
pp. 90-91 ◽  
Author(s):  
V. Beck
Keyword(s):  
Lower Order ◽  
Spherical Aberration ◽  
Mechanical Alignment

Recently a number of experiments have been carried out on a STEM which included a multipole corrector for primary spherical aberration. The results of these experiments indicate that the correction of primary spherical aberration with magnetic multipoles is beset with very serious difficulties related to hysteresis.The STEM and corrector have been described previously. In theory, the corrector should cancel primary spherical aberration so that other aberrations limit the resolution. For this instrument, secondary spherical aberration should limit the resolution to 1 A at 50 kV. A thorough study of misalignment aberrations was made. The result of the study indicates that the octopoles must be aligned to 1000 A. Since mechanical alignment cannot be done to this accuracy, trim coils were built into the corrector in order to achieve the required alignment electrically. The trim coils are arranged to excite all the lower order moments of an element.

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