By Santosh Joshi, Michael Dorff, Indrajit Lahiri

The publication includes thirteen articles, a few of that are survey articles and others study papers. Written by means of eminent mathematicians, those articles have been provided on the foreign Workshop on advanced research and Its functions held at Walchand university of Engineering, Sangli. all of the contributing authors are actively engaged in study fields concerning the subject of the booklet. The workshop provided a finished exposition of the hot advancements in geometric features conception, planar harmonic mappings, whole and meromorphic capabilities and their functions, either theoretical and computational. the new advancements in advanced research and its purposes play a very important function in examine in lots of disciplines.

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Fenn. Ser. A I Math. 425, 1–10 (1968) 24. : Integral means of univalent harmonic maps. Ann. Univ. Mariae Curie-Sklodwska 50, 155–162 (1996) 25. : On the accumulation of the zeros of a Blaschke product at a boundary point. Proc. Am. Math. Soc. 34, 489–496 (1972) 26. : On the fourier series of a step function. Mich. Math. J. 36, 459–475 (1989) 27. : Potential Theory in Modern Function Theory, 2nd Edn. Chelsea Publishing Company, White River (1975) 28. : On the dilatation of univalent harmonic mappings.

1 Z. Boyd and M. Dorff Background In order to explore minimal surfaces more fully, we introduce three important concepts from differential geometry, which is the study of differentiable surfaces in space. For more details on the material from this section, [7]. A surface, M ∈ R3 , can be parametrized by a smooth function x : D → R3 if x(D) = M and x is one-to-one. Parameterizing a surface with smooth functions allows us to do calculus with the surface and gives us a way to translate geometric concepts into rigorous analytic language.

3⎦ 3 The Minimal Surfaces Over the Slanted Half-Planes, Vertical Strips and Single Slit Case 3 For γ = 5π , 4 ⎤ ⎡ −iπ 4 − iπ 4 1 1 e i z−e ⎢1 F (z) = ∓Re ⎣ log + + iπ − iπ − iπ 2 2 4 z+e 4 z+e 4 z + e− 4 Case 4 For γ = ⎥ 2⎦ + c. 7π , 4 ⎡ ⎤ iπ 4 1 z+e 1 1 iπ i ⎢ 1 F (z) = ∓Re ⎣− log + e4 + iπ iπ iπ 4 2 2 z−e 4 z−e 4 z−e 4 Case 5 Let γ ∈ / 53 π 3π 5π 7π , 4, 4, 4 4 ⎥ 2⎦ + c. 4) has simple poles at ieiγ and −ieiγ , and a pole of order 2 at e−iγ . Thus, we may rewrite h (z) as h (z) = A B C D + + + z + ieiγ z − ieiγ z − e−iγ z − e−iγ 2 , where A, B, C, and D can be easily computed using a standard procedure from residue calculus or otherwise.