The aim of this modest study was to shed some light on one of

the useful tools of complex analysis, which is the method of conformal

mapping (Also called conformal transformation). Conformal

transformations are optimal for solving various physical and

engineering problems that are difficult to solve in their original form

and in the given domain. This work starts by introducing the meaning

of a ''Conformal Mapping'', then introducing its basic Properties. In the

second part, it deals with a set of various examples that explain the

behavior of these mappings and show how they map a given domain

from its original form into a simpler one. Some of these examples

mentioned in this study showed that conformal transformations could

be used to determine harmonic functions, that is, to solve Laplace's

equation in two dimensions, which is the equation that governs a variety

of physical phenomena such as the steady-state temperature distribution

in solids, electrostatics and inviscid and irrotational flow (potential

flow). Other mathematical problems are treated. All problems that are

dealt with in this work became easier to solve after using this technique.

In addition, they showed that the harmonicity of a function is preserved

under conformal maps and the forms of the boundary conditions change

accordingly

ABSTRACT In this work, we show the theorem of the convolution of the Abaoub-Shkheam transform and employed it for solving one dimensional linear Volterra integral equations of the second kind.  

In this paper we want to describe the algorithm to find all good starting points for

iteration of 𝑁𝑝(𝑧) to find all the roots of 𝑝(𝑧) by quick and easy way. So in this paper

we have proved that all critical points go to the roots under iteration of Newton’s

function then it will be the good starting points for iteration of 𝑁𝑝(𝑧) .

Abstract-In this paper a new integral transform namely Abaoub-Shkheam transform was introduced. Fundamental properties of this transform were derived and presented such as linearity, change of scale, and first translation or shifting. It is proven and tested to covering equation for temperature distributions in a semi-infinite bar. This transform may solve some different kind of integral and differential equations and it competes with other known transforms like Sumudu and Yang Transform.

Abstract: In this paper, we will use the combination of Regularization method and Direct computation method, or shortly, Regularization-Direct method for solve two dimension- al linear Fredholm integral equations of first kind, by converting the first kind of equation to the second kind by applying the obtain a solution. A few examples are provided to prove the validity and applicability of this approach. regularization method. Then the Direct compotation method is applying to getting the resulting second kind of equation to 

Abstract: In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference. 

In this paper, we first recall the definition of a family of Koenig’s root-

finding algorithms known as Koenig’s algorithms (𝐾௣,௡) for polynomials. In the whole

paper p has degree 𝑑 ≥ 2 with real coefficients and real (and simple) zeros 𝑥௞ , 1 ≤

𝑘 ≤ 𝑑 .

Now we want to discuss Koenig’s algorithms in details where

𝑛 = 4, (𝐾௉,ସ(𝑧)).

 Abstract. We investigate harmonic Bergman spaces bp = bp(Ω), 0 < p < ∞, where Ω = Rn \Zn and prove that bq ⊂ bp for n/(k + 1) ≤ q < p < n/k. In the planar case we prove that bp is non empty for all 0 < p < ∞. Further, for each 0 < p < ∞ there is a non-trivial f ∈ bp tending to zero at infinity at any prescribed rate.

Abstract. We prove that ωu(δ) ≤ Cωf(δ), where u: Ω → Rn is the harmonic extension of a continuous map f: ∂Ω → Rn, if u is a K-quasiregular map and Ω is bounded in Rn with C2 boundary. Here C is a constant depending only on n, ωf and K and ωh denotes the modulus of continuity of h. We also prove a version of this result for Λω-extension domains with c-uniformly perfect boundary and quasiconformal mappings.

In this paper we want to show that each immediate basin of fixed points of Newton’s

function for a real polynomial is simply connected.

Keywords: Newton’s function, Immediate basin, Simply connected.