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Research Article

An Approach to Conformal Transformation Using Symbolic Language Facilities: Application in Electrical Engineering

Algorithm 2

Program for the calculation of the constants which occur in the conformal transformation.
#   The considered function for the calculation of residues will be:
  √ 𝑓 ∢ = 𝐴 β‹… ( 𝑀 βˆ’ π‘Ž ) β‹… ( 𝑀 βˆ’ 𝑏 ) / ( 𝑀 β‹… ( 𝑀 βˆ’ 1 ) ) ;
#   The symbols which do not occur in Algorithm 1, are explained below.
#   The curvilinear integral along the domain contour in direct sense corresponds to a semicircle inside the domain around the point considered singular point clockwise.
#   The residue is the curvilinear integral around the mentioned point divided by 2 πœ‹ 𝐼 .
#   Therefore, the searched value will be obtained by multiplying the residue with πœ‹ 𝐼 with the minus sign before.
#   The residues will be denoted by symbols of two letters, the first will be
𝑅 β€”abbreviation of residue, and the second letter will denote the point, like 𝐺 , for which the residue will be calculated.
#   To calculate the distance between the points 𝐺 and 𝐴 , we shall calculate the circulation along the semicircle at infinity. The corresponding residues at points 𝐺 , 𝐡 , 𝐸 will be:
  𝑅 𝐺 ∢ = r e s i d u e ( 𝑓 , 𝑀 = ∞ + ∞ 𝐼 ) ; 𝑅 𝐡 ∢ = r e s i d u e ( 𝑓 , 𝑀 = 0 ) ; 𝑅 𝐸 ∢ = r e s i d u e ( 𝑓 , 𝑀 = 1 ) ;
#   The corresponding lengths will be:
  𝐿 𝐺 ∢ = 𝐴 πœ‹ 𝐼 ; √ 𝐿 𝐡 ∢ = ( 𝛿 π‘Ž β‹… 𝑏 / πœ‹ ) πœ‹ 𝐼 ; √ 𝐿 𝐸 ∢ = ( 𝛿 βˆ’ 𝑏 βˆ’ π‘Ž + 1 + π‘Ž β‹… 𝑏 / πœ‹ ) πœ‹ 𝐼 ;
#   and there follows:
  √ 𝐿 𝐺 ∢ = 𝛿 𝑏 + π‘Ž + 1 βˆ’ π‘Ž β‹… 𝑏 ;
#   But according to Figure 1:
  𝐿 𝐺 ∢ = 𝛿 𝐼 ;  𝐿 𝐡 ∢ = 𝛿 𝐼 ;  √ 𝐿 𝐸 ∢ = 𝛿 𝑏 + π‘Ž βˆ’ 1 βˆ’ π‘Ž β‹… 𝑏 ;  𝐿 𝐸 ∢ = 𝑏 0 ;
#   From the pair of relations 𝐿 𝐡 , there follows that the the product ( π‘Ž β‹… 𝑏 ) is equal to unity.
#   There follows:
# ( 𝑏 βˆ’ 1 ) 2 / 𝑏 = ( 𝑏 0 / 𝛿 ) 2 and 𝑏 0 √ / 𝛿 = ( 𝑏 βˆ’ 1 ) / 𝑏 .
#   Using the details above, we obtain the relations of (6.2) and (6.3), where instead of 𝑏 0 / 𝛿 we put simply 𝑏 0 , namely the relative length, expressed as a ratio to base 𝛿 .