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A flexible cable suspended between two points at the same height naturally forms a curve known as a catenary. This shape results from the balance between the cable’s weight and the tension acting along its length, representing a state of mechanical equilibrium. Unlike simpler approximations, the true shape of a hanging cable is described using hyperbolic functions.Hyperbolic functions are closely related to exponential functions and are named for their connection to the geometry of the...
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An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...
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The shape of a suspension bridge cable hanging under its own weight is described by a catenary curve, which is modeled using the hyperbolic cosine function. This mathematical model accurately captures the balance between gravity and tension acting along the cable. When a particular vertical position on the cable is known, the corresponding horizontal position can be determined using the inverse hyperbolic cosine function, allowing for a detailed analysis of the cable's geometry.Inverse...
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A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...
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Trigonometric functions exhibit periodic and symmetrical behavior, deeply rooted in the unit circle. The sine and cosine functions correspond to the vertical and horizontal projections, respectively, of a point rotating counterclockwise around the circle. These functions trace smooth, repeating waveforms with identical periods and bounded ranges. The tangent function is defined as the ratio of sine to cosine and produces an unbounded curve that repeats every units, with vertical asymptotes...
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On certain subclass of meromorphic spirallike functions involving the hypergeometric function.

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We introduce a new subclass of meromorphic spirallike functions, denoted M(m)(1)(θ, λ, η). This study establishes integral representations, convolution properties, and coefficient estimates for this new class.

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Area of Science:

  • Complex Analysis
  • Geometric Function Theory

Background:

  • Meromorphic functions are essential in complex analysis.
  • Spirallike functions exhibit unique geometric properties.

Purpose of the Study:

  • Introduce and investigate a novel subclass of meromorphic spirallike functions: M(m)(1)(θ, λ, η).
  • Extend existing theories in geometric function theory.

Main Methods:

  • Utilizing techniques from complex analysis.
  • Applying methods for integral representations.
  • Investigating convolution properties.
  • Deriving coefficient estimates.

Main Results:

  • Established integral representations for the new function subclass.
  • Proved key convolution properties.
  • Obtained coefficient estimates for M(m)(1)(θ, λ, η) functions.
  • Extended findings from prior research.

Conclusions:

  • The study successfully defined and analyzed the M(m)(1)(θ, λ, η) subclass.
  • The derived results offer significant extensions to the field of meromorphic spirallike functions.