J. Nig. Soc. Phys. Sci. 5 (2023) 1392 Journal of the Nigerian Society of Physical Sciences Pre-functions and Extended pre-functions of Complex Variables A. Thirumalaia, K. Muthunagaia,∗, Ritu Agarwalb aSchool of Advanced Sciences, VIT University, Chennai-600 127, Tamil Nadu, India bDepartment of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, India Abstract Pre-functions are functions that possess a sequence { fn(z,β)} which tends to one of the elementary functions as n tends to infinity and β tends to 0. The main objective of this paper is to broaden the scope of pre-functions from functions of a real variable to functions of a complex variable by introducing pre-functions of a complex variable. We have analyzed the pre-functions of a complex variable for their properties. The pre-Laguerre, pre-Bessel and pre-Legendre polynomials of a complex variable have been obtained as special cases. Graphs have been used to visualize complex pre-functions. DOI:10.46481/jnsps.2023.1427 Keywords: pre-exponential functions, pre-trigonometric functions, pre-hyperbolic functions, Extended pre-functions. Article History : Received: 01 March 2023 Received in revised form: 09 May 2023 Accepted for publication: 10 May 2023 Published: 21 May 2023 c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: O. Adeyeye 1. Introduction Exponential and logarithmic functions have wide variety of ap- plications in science, medicine, business and many other fields. Exponential functions are used to describe growth or decay of a quantity whose rate of change has a relation to its present value. Logarithms have the ability to measure quantities that are vastly different but need an easy way to be talked about and compared to. On the other hand, trigonometry can be used in music, roofing a house, cartography, satellite system in naval, aviation industries and in many other fields. Deo and Howell [1] introduced and studied trigonometry and trigonometry like functions. Khandeparkar et al. [2] have introduced and studied pre-functions of a real variable. Moti- vated by their work, we have defined pre-functions of a com- ∗Corresponding author tel. no: +91 9840084991 Email address: muthunagai@vit.ac.in (K. Muthunagai ) plex variable. Functions which possess a sequence { fn(z,β), z ∈ C,β ≥ 0} are called pre-functions of a complex variable z, if they tend to one of the elementary functions as n → ∞ and β → 0. Pre-functions also possess some of the properties possessed by the elementary functions but not properties like periodicity. Pre-functions were found to be very simple and useful in the study of differential equations. For the methods and solutions of various differential equations one can refer [3- 10]. 2. The pre-exponential function of a complex variable Exponential functions play a significant role in almost every branch of Mathematics. In this section, we have determined a set of functions called pre-exponential functions, owning a sequence that generalizes the exponential function ex p(z). 1 Thirumalai et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 2 For any complex number z, the series form of pre- exponential function is given by, pexp(z,β) = 1 + z1+β Γ(2 + β) + z2+β Γ(3 + β) + z3+β Γ(4 + β) + .... = 1 + ∞∑ n=1 zn+β Γ(n + 1 + β) , β ≥ 0, (1) β being the parameter. Figure 1: Graphs of pexp(z, 0), pexp(z, 1) and pexp(z, 2) Figure 1 represents the behaviour of the pre-exponential functions pexp(z, 0), pexp(z, 1) and pexp(z, 2) in order. Note that when β = 0, pexp(z, 0) = exp(z). In general, pexp(z, n) = ∗pexp(z, n − 1) − zn n! = pexp(z, n − 2) − zn−1 (n − 1)! ..., n = 1, 2, 3, .. Figure 2: Z-Plane Specifically, pexp(z, 1) = exp(z) − z pexp(z, 2) = exp(z) − z − z2 2! pexp(z, 3) = exp(z) − z − z2 2! − z3 3! = exp(z) − S 3 where S 3 is the partial sum of pexp(z, 3). For n ∈ N, pexp(z, n) = exp(z) − S n (2) where S n = ∑n r=1 zr r! . Also, note that pexp(z, n) = 1, ∀ z ∈ C as n →∞. Replacing z by −z in (1), we have, pexp(−z,β) = 1 − (−1)β { z1+β Γ(2 + β) − z2+β Γ(3 + β) + z3+β Γ(4 + β) − ... } = 1 + (−1)β ∞∑ n=1 (−1)n zn+β Γ(n + 1 + β) . (3) In (3), replacing β by 0 pexp(−z, 0) = 1 − z 1! + z2 2! − z3 3! + ... = exp(−z) (4) In short, pexp(−z, n) = exp(−z) − S n where S n = ∑n r=1(−1) r zr r! . 3. Pre-trigonometric Functions of a Complex Variable The pre-trigonometric functions of a complex variable are defined by Y1(z,β) = pcos(z,β) = 1 − z2+β Γ(3 + β) + z4+β Γ(5 + β) − z6+β Γ(7 + β) + ... = 1 + ∞∑ n=1 (−1)n z2n+β Γ(2n + 1 + β) , z ∈ C, β ≥ 0 (5) and Y2(z,β) = psin(z,β) = z1+β Γ(2 + β) − z3+β Γ(4 + β) + z5+β Γ(6 + β) − ... = ∞∑ n=0 (−1)n z2n+1+β Γ(2n + 2 + β) , z ∈ C, β ≥ 0. (6) Table 1 gives the expressions for the pre-trigonometric func- tions when β takes the values 0,1,2 and 3. 2 Thirumalai et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 3 Figure 3: W-Plane Figure 4: pcos(z, 1) = 1 − z 3 3! + z5 5! − z7 7! Table 1: Expressions for the pre-trigonometric functions S.No (z,β) pcos(z,β) psin(z,β) 1 (z, 0) cos(z) sin(z) 2 (z, 1) sin z − z + 1 1 − cos z 3 (z, 2) −cosz − z 2 2 + 2 z − sin z 4 (z, 3) −sin z + z − z 3 6 + 1 cos z + z2 2 − 1 4. Euler’s Formula For a complex number z, exp(iz) = cos z + i sin z. (7) Using (1), the above expression can be rewritten as pexp(iz,β) = 1 − (i)β { z2+β Γ(3 + β) − z4+β Γ(5 + β) + z6+β Γ(7 + β) + ... } +i(i)β { z1+β Γ(2 + β) − z3+β Γ(4 + β) + z5+β Γ(6 + β) − ... } = 1 − (i)β{1 − pcos(z,β) − i psin(z,β)}, β ≥ 0 (8) Figure 5: pcos(z, 0.6) = 1 − z 2.6 Γ(3.6) + z4.6 Γ(5.6) − z6.6 Γ(7.6) Figure 6: psin(z, 1) = z 2 2! − z4 4! − z6 6! Figure 7: psin(z, 0.6) = z 1.6 Γ(2.6) − z3.6 Γ(4.6) − z5.6 Γ(6.6) Figure 8: pcosh(z, 1) = 1 + z 3 3! + z5 5! + z7 7! which is the general form of Euler’s Formula for pre- exponential function. Clearly eiz = pexp(iz, 0) = pcos(z, 0) + i psin(z, 0) = cos z + i sin z. Using −iz in place of iz, we have pexp(−iz,β) = 1 − (i)β{1 − pcos(z,β) + i psin(z,β)}, (9) which results in e−iz = cos z − i sin z, when β = 0. 3 Thirumalai et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 4 Figure 9: pcosh(z, 0.7) = 1 + z 2.7 Γ(3.7) + z4.7 Γ(5.7) + z6.7 Γ(7.7) Figure 10: psinh(z, 1) = z 2 2! + z4 4! + z6 6! Figure 11: psinh(z, 0.7) = z 1.6 Γ(2.6) − z3.6 Γ(4.6) − z5.6 Γ(6.6) Figure 12: M3,0(z, 1) = 1 − z4 4! + z7 7! − z1 0 10! 5. Relation between pre-circular and pre- exponential func- tions Using Euler’s formula, the relation between circular and ex- ponential functions, we arrive at the following: pcos(z,β) = (−i)β pexp(iz,β) + (i)β pexp(−iz,β) 2 − (i)β + (−i)β 2 + 1 psin(z,β) = (−i)β pexp(iz,β) − (i)β pexp(−iz,β) 2i − (−i)β − (i)β 2i . ptan(z,β) = i (−i)β − (i)β − (−i)β pexp(iz,β) + (i)β pexp(−iz,β) (−i)β pexp(iz,β) + (i)β pexp(−iz,β) − (i)β − (−i)β + 2 psec(z,β) = 2 (−i)β pexp(iz,β) + (i)β pexp(−iz,β) − (i)β − (−i)β + 2 Table 2: Expressions for the pre-hyperbolic functions S.No (z,β) pcosh(z,β) psinh(z,β) 1 (z, 0) cosh(z) sinh(z) 2 (z,1) sinh z − z + 1 cosh z − 1 3 (z,2) cosh z − z 2 2 sinh z − z 4 (z, 2n) cosh z − ∑n r=1 z2r (2r)! sinh z − ∑n r=1 z2r−1 (2r−1)! pcosec(z,β) = 2i (−i)β pexp(iz,β) − (i)β pexp(−iz,β) + (i)β − (−i)β pcot(z,β) = −i (−i)β pexp(iz,β) + (i)β pexp(−iz,β) − (i)β − (−i)β + 2 (−i)β − (i)β − (−i)β pexp(iz,β) + (i)β pexp(−iz,β) (10) whenever they exist. For β = 0, these results give the relation between circular functions and exponential functions. 6. Pre-hyperbolic Functions of a Complex Variable The pre-hyperbolic sine and cosine functions are defined by H1(z,β) = pcosh(z,β) = 1 + z2+β Γ(3 + β) + z4+β Γ(5 + β) + z6+β Γ(7 + β) + ... = 1 + ∞∑ n=1 z2n+β Γ(2n + 1 + β) , z ∈ C, (11) and H2(z,β) = psinh(z,β) = z1+β Γ(2 + β) + z3+β Γ(4 + β) + z5+β Γ(6 + β) + ... = ∞∑ n=0 z2n+1+β Γ(2n + 2 + β) , z ∈ C. (12) Table 2 gives the expressions for the pre-hyperbolic functions when β takes the values 0, 1, 2 and 2n. 7. Relation between pre-hyperbolic and pre- exponential functions pcosh(z,β) = (−1)β pexp(z,β) + pexp(−z,β) 2 − 1 − (−1)β 2 , psinh(z,β) = (−1)β pexp(z,β) − pexp(−z,β) 2 − (−1)β − 1 2 , ptanh(z,β) = (−1)β pexp(z,β) − pexp(−z,β) − (−1)β + 1 (−1)β pexp(z,β) + pexp(−z,β) − 1 + (−1)β , psech(z,β) = 2 (−1)β pexp(z,β) + pexp(−z,β) − 1 + (−1)β pcosech(z,β) = 2 (−1)β pexp(z,β) − pexp(−z,β) − (−1)β + 1 , pcoth(z,β) = (−1)β pexp(z,β) + pexp(−z,β) − 1 + (−1)β (−1)β pexp(z,β) − pexp(−z,β) − (−1)β + 1 , if exists. Assigning β the value 0, we find these relations reduc- ing to the relations between Exponential and Hyperbolic func- tions. 4 Thirumalai et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 5 8. Extended pre-functions of a Complex Variable From generalized pre-trigonometric and pre-hyperbolic functions of a complex variable we have, pexp(−z,β) = 1 − (−1)β { z1+β Γ(2 + β) − z2+β Γ(3 + β) + z3+β Γ(4 + β) − .... } = 1 + (−1)β ∞∑ n=1 (−1)n zn+β Γ(n + 1 + β) . Trisection of the above series leads us to three infinite abso- lutely convergent series for z ∈ C and β ≥ 0. They are M3,0(z,β) = 1 + ∞∑ n=1 (−1)n z3n+β Γ(3n + 1 + β) M3,1(z,β) = ∞∑ n=0 (−1)n z3n+1+β Γ(3n + 2 + β) M3,2(z,β) = ∞∑ n=0 (−1)n z3n+2+β Γ(3n + 3 + β) , (13) with the initial conditions M3,0(0,β) = 1, M3,1(0,β) = 0, M3,2(0,β) = 0. One can easily verify M′3,0(z,β) = −M3,2(z,β) M′3,1(z,β) = (−1) β z β Γ(1 + β) + M3,0(z,β) − 1 M′3,2(z,β) = M3,1(z,β). Rewriting the above system in matrix form, we have  M3,0(z,β) M3,1(z,β) M3,2(z,β)  ′ =  0 0 −1 1 0 0 0 1 0   M3,0(z,β) M3,1(z,β) M3,2(z,β)  +  0 (−1)β z β Γ(z+β) − 1 0   M3,0(z,β) M3,1(z,β) M3,2(z,β)  =  1 0 0  (14) We find the infinite series represented by M3,0(0,β) = 1, M3,1(0,β) = 0, M3,2(0,β) = 0 to be the solutions of the sys- tem of non-homogeneous equations given by (12). The expres- sions in (11) define the extended pre-trigonometric functions for n = 3 . Specifically when β = 1 , we have M3,0(z, 1) = 1 + ∞∑ n=1 (−1)n z3n+1 Γ(3n + 2) = M3,1(z, 0) − z + 1 M3,1(z, 1) = ∞∑ n=0 (−1)n z3n+2 Γ(3n + 3) = M3,2(z, 0) M3,2(z, 1) = ∞∑ n=0 (−1)n z3n+3 Γ(3n + 4) = −M3,0(z, 0) + 1 (15) Figure 13: M3,0(z, 0.5) = 1 − z3.5 Γ(4.5) + z6.5 Γ(7.5) − z10.5 Γ(11.5) Figure 14: M3,1(z, 1) = z2 2! − z5 5! + z8 8! Figure 15: M3,1(z, 0.5) = z1.5 Γ(2.5) − z4.5 Γ(5.5) − z7.5 Γ(8.5) Figure 16: M3,2(z, 1) = z3 3! − z6 6! + z9 9! 9. Absolute Convergence, analyticity and univalence of pre- functions We know that every absolute convergent series is conver- gent. But the converse is not true. In this section we have dis- cussed about the absolute convergence of pre-exponential func- tion using Ratio test. pexp(z,β) = 1 + ∞∑ n=1 zn+β Γ(n + 1 + β) 5 Thirumalai et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 6 Figure 17: M3,2(z, 0.5) = z2.5 Γ(3.5) − z5.5 Γ(6.5) − z8.5 Γ(9.5) Figure 18: pexp(z, 0.8) = 1 + z 1.8 Γ(2.8) + z2.8 Γ(3.8) + z3.8 Γ(4.8) Consider ∣∣∣∣∣ an+1an ∣∣∣∣∣ = ∣∣∣∣∣ Γ(n + 1 + β)Γ(n + 2 + β) ∣∣∣∣∣ = ∣∣∣∣∣ (n + β)!(n + 1 + β)! ∣∣∣∣∣ = ∣∣∣∣∣ (n + β)!(n + β + 1)(n + β)! ∣∣∣∣∣ = ∣∣∣∣∣ 1n + β + 1 ∣∣∣∣∣ lim n→∞ ∣∣∣∣∣ an+1an ∣∣∣∣∣ = 0 In similar lines the absolute convergence of the other pre- functions can also be proved. As the pre-exponential, pre- trigonometric, pre-hyperbolic and extended pre-functions are all polynomials with infinite number of terms, they are analytic throughout the complex plane, (i.e.) they are entire functions. Also they are univalent. As any function that is both analytic and univalent is conformal, so are pre-functions. 10. The transformation w = 1pexp( z,β) In this section, we have obtained the image of |pexp(z,β) − 2| = 1 under the transformation w = 1pexp(z,β) . w = 1pexp(z,β) ⇒ pexp(z,β) = 1 w and pexp(z,β) = 1 + ∑ ∞ n=1 zn+β (n+β)! . |pexp(z,β) − 2| = 1 ⇒ ∣∣∣∣∣1 + ∞∑ n=1 zn+β (n + β)! − 2 ∣∣∣∣∣ = ∣∣∣∣∣ 1w − 2 ∣∣∣∣∣ ⇒ ∣∣∣∣∣ ∞∑ n=1 zn+β (n + β)! − 1 ∣∣∣∣∣ = ∣∣∣∣∣ 1 − 2ww ∣∣∣∣∣ Approximating the series to only one term we have∣∣∣∣∣ z22 − 1 ∣∣∣∣∣ = ∣∣∣∣∣ 1 − 2ww ∣∣∣∣∣ [n = 1,β = 1] Now ∣∣∣∣∣ 1 − 2ww ∣∣∣∣∣ = 1∣∣∣∣∣1 − 2w ∣∣∣∣∣ = |w| |1 − 2(u + iv)| = |u + iv| (u − 2 3 )2 + v2 − 1 9 = 0 which is a circle in the w-plane. Figures 2 and 3 are visualiza- tion of the given transformation. 11. Visualization of Certain pre-functions The extended trigonometric functions M3,0(z), M3,1(z) and M3,2(z) are found to be the linear independent solutions of the differential equation z ′′′ + z = 0. The properties possessed by these functions are similar to that of the classical trigonometric functions but for periodicity. Due to lack of periodicity we see the graph to be oscillating with interlacing zeros. The paramet- ric equations y1 = M3,0(z), y2 = M3,1(z), y3 = M3,2(z) will generate a surface y31 − y 3 2 + y 3 3 + 3y1y2y3 = r 3. For β = 1, the graphs of pre-trigonometric and extended pre-trigonometric functions are found to be oscillating and at the same time loos- ing periodicity. The graphs in Figures 4-18 show how specific pre-functions behave for some fixed values of β. 11.1. Special Cases Following are some of the special cases obtained as a result of our study about the pre-functions of a complex variable. 1. From the first identity of (13), we obtain M3,0(z1 + z2, 1) = M3,1(z1 + z2, 0)−(z1 + z2) + 1(16) by replacing z by z1 + z2 in it. 2. Trisecting the series (1) for pexp(z,β), three infinite absolutely convergent series namely N3,0(z,β), N3,1(z,β), N3,2(z,β) for z ∈ C, β ≥ 0 have been obtained and these series define extended hyper- bolic functions for n = 3. Proceeding in similar lines as it has been done for n = 2 and n = 3, n-section of the infinite series pexp(−z,β) and pexp(z,β) give rise to generalized extended trigonometric and hyperbolic functions. 6 Thirumalai et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 7 3. Generating function for pre-Laguerre polynomial can be obtained from pexp(z,β), by replacing z using −zyz−1 . 1 (1 − z) pexp ( −zy 1 − z ,β ) = 1 (1 − z) { 1 + (−1)β ∞∑ r=0 (−1)r zr+βyr+β (1 − z)r+βΓ(r + 1 + β) } = 1 (1 − z) + ∞∑ r=0 (−1)r+β Γ(r + 1 + β) zr+βyr+β (1 − z)r+β+1 = 1 (1 − z) + ∞∑ r=0 (−1)r+β Γ(r + 1 + β) zr+βyr+β(1 − z)−(r+β+1) = 1 (1 − z) + ∞∑ r=0 (−1)r+β Γ(r + 1 + β) zr+βyr+β ∞∑ t=0 (r + t + β)! (r + β)!t! zt = 1 (1 − z) + ∞∑ r,t=0 (−1)r+β (r + t + β)! Γ(r + β + 1)(r + β)!t! yr+β ∞∑ t=0 zr+t+β For a fixed value of r and taking r + t = n, the coefficient of zn is (−1)r+β (n + β)! Γ(r + β + 1)(r + β)!(n − r)! yr+β. Taking all possible values of r into account, the total co- efficient of zn is obtained to be n∑ r=0 (−1)r+β (n + β)! Γ(r + β + 1)(r + β)!(n − r)! yr+β = Ln(y,β), s = n − r ≥ 0 or r ≤ n. Here Ln(y,β) represents the Laguerre Polynomial when β = 0. Ln(y, 0) = n∑ r=0 (−1)r n! (r)!(n − r)! yr = Ln(y) 1 (1 − z) pexp {( −zy 1 − z ,β ) − 1 } = ∞∑ n=0 zn+βLn(y,β) 4. We can also obtain the generating function for pre- Bessel polynomial using pre-hyperbolic sine function. In psinh(z,β), replacing z by zx2 , we have (3n + 1 + β)! −(n + 1 −β)! psinh ( zx 2 ,β ) = (3n + 1 + β)! −(n + 1 −β)! { ∞∑ n=0 z2n+1+βx2n+1+β 22n+1+βΓ(2n + 2 + β) } = ∞∑ n=0 (3n + 1 + β)!z2n+1+βx2n+1+β −(n + 1 −β)!22n+1+βΓ(2n + 1 + β) = ∞∑ n=0 (3n + 1 + β)!z2n+1+βx2n+1+β −(n + 1 −β)!22n+1+β(2n + 1 + β)! For a fixed n and setting k = 2n + 1, the coefficient of zn is n∑ k=0 (k + n + β)! (n − k + β)!(k + β)! xk+β 2k+β = Yn(x,β), k ≤ n. Here Yn(x,β) represents the Bessel Polynomial when β = 0. Yn(x, 0) = n∑ k=0 (k + n)! (n − k)!(k)! ( x 2 )k = Yn(x) (3n + 1 + β)! −(n + 1 −β)! psinh ( zx 2 ,β ) = ∞∑ n=0 zn+βYn(x,β) 5. Replacement of z using zx2 yields the generating function for pre-Legendre polynomial (3n + α + 1)! (3m + n + 1 + α)!(2n + m + 1 + α)! psin ( zx 2 ,α ) = (3n + α + 1)! (3m + n + 1 + α)!(2n + m + 1 + α)! ∗ { ∞∑ n=0 (−1)n z2n+1+α x2n+1+α 22n+1+αΓ(2n + 2 + α) } = ∞∑ n=0 (−1)n(3n + α + 1)!z2n+1+α x2n+1+α (3m + n + 1 + α)!(2n + m + 1 + α)!22n+1+αΓ(2n + 1 + α) = ∞∑ n=0 (−1)n(3n + α + 1)!z2n+1+α x2n+1+α (3m + n + 1 + α)!(2n + m + 1 + α)!22n+1+α(2n + 1 + α)! (17) Fixing n and taking α = −(n + 2m + 1−β), the coefficient of zn is M∑ m=0 (−1)m+β (2n − 2m + β)!xn−2m+β (m + β)!(n − m + β)!(2n−2m+β)(n − 2m + β)! = Pn(x,β), (18) m ≤ n. Here Pn(x,β) represents the Legendre polynomial when β = 0. Pn(x, 0) = M∑ m=0 (−1)m (2n − 2m)!xn−2m m!(n − m)!(2n−2m)(n − 2m)! = Pn(x) =⇒ (3n + α + 1)! (3m + n + 1 + α)!(2n + m + 1 + α)! psin ( zx 2 ,α ) = ∞∑ n=0 zn−2m+βPn(x,β) (19) 12. Conclusion In this paper, we have introduced and investigated the prop- erties of pre-functions and extended pre-functions for a com- plex variable. By fixing the value of β, we were able to graph these pre-functions and extended pre-functions. For suit- able choices of z, we observe that these functions reducing to Leguerre, Bessel, and Legendre polynomials, among other spe- cial functions. Some more special functions can be derived by assuming simple functions for the variable z. References [1] S. G. Deo & G. W. Howell,”A highway to trigonometry”, Bull.Marathawada Mathematical society 1 (2000) 26-62. [2] R.B. Khandeparkar, S. Deo & D. B. Dhaigude, “pre-exponential and pre-trigonometric Functions”, Communications in Applied Analysis 14 (2010) 99. 7 Thirumalai et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 8 [3] V. O. Atabo & S. O. 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