2 Analyticity of logarithmic Mittag-Leffler function ζ _m ^(α)(z)

To determine foe malyticity of logarithmic Mittag -Leffler function ζ _m ^(α)(z), we present following lemmas and theorems

Lemma 2.1. If s Є C and α Є R, α > 0, then for 0 < c < 1, the Mellin - Barnes type integral of Eα is given by

Proof The contour integral m (2.1) separates all potes s = -k.k.= 0,1,2,... to the left and all poles s = n +1 , n = 0,1,2,... to the right. Hence, by calculus of residue and the sum of the residue, at the points s= 0, -1, 2, ... fo the contour integral in (2. 1), we write

Now, using the definition of E _α (z) given in (1.1) with z = 1, (2.2) becomes

Hence, the result (2.1) is found. This result is obviously identical to the formula (1.5) (when, z = 1)

Set α = 1 m the formula (2.1) to get the Mellim - Barnes type integral for E₁ e = as

Lemma 2.2. If f (z) is holomorphic on a simply connected domain ⅅ ⊂ ℂ and 0 ∉ f (ⅅ) then there exists another function(z) holomorphic on ⅅ and f (ⅅ) = ℂ∖{0} is not simply connected .

Proof. By the relation g(z)=log f (z), we may write f (z)=exp[g(z)]=exp[g(z)+2nΠi] ∀n= 0,1,2, . . . Since there is no complex number [g(z)+ 2nΠi] ∈ ℂ ∀n = 0,1,2, . . . so that f (z) becomes zero, hence 0 ∉ f (ℂ) ∀ z ∈ ⅅ = ℂ. Again, ⅅ is simply connected, then, there exists Thus we have

Now in (2.5), for example set f (z) = e ^z , then by the relation g(z) = log f (z), the left hand side of (2.5) becomes z, and that of right hand side is z - z₀ and thus ultimately, the result is C0ntradlctl0n. Hence g(z) could not be defined for a continuous single-valued branch of log f (z). Therefore in this case f (ⅅ) = ℂ \{0} is not simply C0nneCted. □

Lemma 2.3. If f (z) is an entire function ∀z ∈ ℂ, in the complex plane, then there exists another function G(z) = e ^-z log f (z) analyticyz ∀z ∈ ℂ in that complex plane.

Proof By the relation G(z) = e ^-z log f (z), we have log f (z) = e ^z G(z) s0 that by Lemma 2.2, we get

Now for trie complex: anylysis involved, we refeo ^{Conway (Conway, 1973}). It eollows immediately that the proof. of the Lemma 2.33 is triviel. To verify the equality, we set /(z) =s ef th lbotls sides oft this result and the Lemma is followed.

For another example, set f (z)= , in Lemma 2.3, then we have an analytic function

Now on differentiating both the sides of (2.7), we have

which is a linear differential equation

whence, upon integration, we obtain an integral function

C is arbitrary constant, or its equivalent form

C is arbitrary constant.

But, by (2.7) G(0) = 0 and lim_z→0 log . Thus C = 0 and we find that

Now multiply both sides of (2.8) by A ^(α) and then use(1.1) and (1.2) to get a valuable relation

where

Again then, letting in (2.9),we have

and

where the equation = 0 has a single root ξ = z in a complex ξ - plane in a region ⅅ. □

Theorem 2.1. If , and X ^m(α) (z) are given in (1.1) and (2.9), respectively, ∀z ∈ ⅅ- ∂ⅅ ⊂ ℂ, α∈ ℝ α>0, m ℕ* ℕ*= {2,3,4…, L} < ∞, and there is a continuous function G(ξ,z)= lim _ξ→z the equation in a region ⅅ ofα complex ξ -plane , has single root ξ = z, and on a regular arc ∂ⅅ of length l, where ν _k is a point on the regular arc ∂ⅅ lying between the points ξ_k-1 and ξ_k and v _k = ξ (Τ _k ) = x(Tk ) + iy(Tk) ∀ _tk-1 ≤ T _k ≤ tk ; k = 1,2,..., also then there exists an dbsoliite value

Proof. Since in the Theorem 2.1, ∀_z ∈ ⅅ- ∂ⅅ ⊂ ℂ α ∈ ℝ α >0, m ∈ ℕ*, ℕ*= {2,3,4…,} L<∞ the equation , in a region ⅅ of a complex ξ - plane , has single root ξ = z on a regular arc ∂ ⅅ of length l, where v _k is a point on the regular arc ∂ⅅ lying between the points ξ _k-1 and ξ _k and v _k = ξ (T _k )= x(T _k ) + iy(Tk) ∀ _tk-1 ≤ T _k ≤ t _k ;k = 1_,2,…, so let the points α = ξ₀ < ξ ₁ <…< ξ _n = b lie on the arc of length l in the interval (a,b) ⊂ (-∞,∞ ). Then by (2.10) and by ^{Campos theory for comple_X derivatives (Campos, 1984}), the left hand side of the integral of the equation (2.11) is written as

Hence, the theorem is followed.

Full proof of the inequality

Consider

By Eqns. (2.9) and (2.10), Such that then we find

And then, we get

Theorem 2.2. Iff: ℝ → ℝ de/med fry the formula T ( f ) ,dξ then,for there exists anestimate

Proof. Statmg with the inequality given in and the inequality of the integral (2.11), and using the techniques for Cauchy Schwarz ine quality in (^{Steele, 2004}, p.108), the left hand(2.12) is written as

On applying the Theorem 2.1, we have

In this way, we obtain the estimate

Hence, inequality of the Theorem 2.2 is established.

3 Campos theory and extended derivative of complex function f (z) with respect to a complex function Ω.m(α) (z)

In this section, we introduce the theory and concept of derivative of complex order defined by ^{Campos (Campos, 1984}, Eqn. (1), p. 113) as the derivative of complex order v of complex function /(z) with respect to z, the complex variable, in the form

Here in formula (3.1), v is any complex number other than a negative integer, implying that the Gamma function Γ(v + 1) is analytic, as its poles; v + 1 = 0, - 1, - 2,... ; have been excluded. The function f (z) is a complex function of a complex variable z, analytic in the (open) interior ⅅ - ∂ⅅ of the (closed) region of mtegration ⅅ, which may or may not include the point-at-infinity; the borradary ∂ⅅ is a simple and regular (or rectifiable) curve, and on it the function f (ξ) satisfies one of the progressrvdy less stringent conditions that it is analytic and it is unifomily Contuous f (z) → f (ξ) as z i0 ⅅ - ∂ⅅ tends to ξ on ⅅ; it is bounded on ∂ⅅ, with at most a finite number of discontinuities.

The Cauchy kernel (ξ −z)−v−1 is single valued if v is an integer, and the principal branch of this many-valued function is taken if v is not an integer. The contour of integration ∂ⅅ is described in the positive (counterclockwise) direction around z, and does not cross any branch-cuts in the ξ-plane; if any of the branch-cuts is infinite, the contour ∂ⅅ must be open, and can only touch the cut at the point at infinity; if there are no branch-cuts, or all are finite, the contour ∂ⅅ is closed and finite (i.e. a loop), and may or may not touch a branchpoint, as necessary to ensure that the integrand returns to the initial value after describing the loop; the integral should be independent of the contour (provided it is deformed without crossing branch-cuts or touching any new branch-points and z remains in its interior), and uniformly convergent with regard to z.

In a case, when we set v = +n, ∀n∈ ℤ +, in (3.1), it becomes the Cauchy (1825) integral theorem, given by Campos (Campos, 19884, Eqn. (2), p. 114)

(See also, (^{Conway, 1973}), (^{Oldham & Spanier, 2006}) and (^{Pathan & Kumar, 2019})).

In another case, when a function f (z) with a branch - point at z = z0 and with complex exponent μ, not an integer, that is f (z) = (z−z0)μ g(z), g(z)is analytic, then the derivative of complex order v of complex function f (z) of complex variable z, is defined by an Pochhammer’s type integral as (see ^{Campos (Campos, 1984}, Eqn. (13), p. 118))

Theorem3.1. IfG(ξ,z) = and the equation has single root ξ = z in ⅅ, Ω(α)m (z) is given (2.10), and the function f : ⅅ → ℂ, defined by f (z) = (z- z0)μ g(z) ∀z,zo ∈ ℂ, g(z) is analytic in complex ξ -plane, then ∀v,μ ∈ ℂ there exists a extended Pochhammer’s type contour integral formula for complex _ order derivative with respect to the function Ω(α)m (z) as

Proof. By the rulings of (3.1) and (3.3), we define

Then, in it apply the results due to (2.10), the Theorem 2.1 and (1.1).Thus, we get the result

By making use of this result, we easily obtain (3.4). This completes the proof of the Theorem.

Remark 3.1. To verify above result (3.4), put α = 1 in formula (3.4) and use the property of Eα in (1.3) for logee Eα as l logee Eα |α = 1 =1, then A(α) =1 and Ω(α)m (z)=z and hence, this formula becomes equivalent to the formula (3.3), (see ^{Campos (Campos, 1984})).

4 An application to obtain the extended Pochhammer's type integral representations and Rodrigues formulae of hypergeometric functions

In this section, we apply our result (3.4) of the Theorem 3.1 and obtain the extended Pochhamrne°'s type integral representations and Rodrigues formulae of vaious known hy-p°rgeometric fonctions.

To achieve our goal, we present an integral representation of Kummer's function 1 F1[.] by Erdelyi (^{ErdImagelyi, Magnus, Oberhettinger, & Tricomi, 1953}, Vol. 1, p. 271) due to ^{Campos (Campos, 1984}, Eqn. (37) with (36b)) as the Pochhammer type contour integral given by

where, B =

Now in (3.4), put μ = b -1,v = b- c,z0 = 0,g(n) = ez, the nomplex b - 1 is not an integer, and thus multiply both of the sides by the function Γ(1 - b) Γ (c)z1-c and use the results (2.10) and (1.1), and compare with (4.1) to get

Therefore, the extended Pochhammer type contour integral representation of Kummer's type hypergeometric function F1 [b;c;z] is of the form

Remark 4.1. Setting α = I in the result (4.3), we find the results of Compos (^{Campos, 1984}, Eqn. (37) with (36b)).

Again in (3.4), set μ = b - 1, v = b - c,zo = 0,g(z) = (1 - σz)-Y(1 - ρz)-β, the complex b - 1 is not an integer, │σz│< 1, │ρz│ < 1 and then multiply both the sides by the function Γ(1 - b)Γ(c)z1-c . Now use the results of (4.2) together with the formula (^{Srivastava and Manocha (H. M. Srivastava & Manocha, 1984}1, p.290)) to obtain the Pochhammer contour integral of App ell's function F1 [.,.] of two variables, given by

Further, put σ = 1, ρ = 0 in Eqn. (4.4), to find the Pochhammer's type contour integral formula for the Gaussian hypergeometric function 2 F1 [.], given by

Remark 4.2. Setting α = 1 in the result (4.5), we find the results of Compose (^{Campos, 1984}, Eqn. (32) with (31b)).

5 An application in the case of spread of infectious disease in terms of multiple hypergeometric functions

In this section, we consider another property of the logarithmic Mittag - Leffler function to obtain the area of spread of infectious disease for different parameters and weight functions. The results are evaluated in terms of Lauricella's multiple hypergeometric functions.

Theorem 5.1. IfG(z) = (given in Lemma 2.3) and there exists a function F : ℝ → ℝ, defined and then

Proof. Since G(z) = , we may write

so that on using the formula (see ^{Mathai and Haubold (Mathai & Haubold, 2008}, p. 84)),

we may further write as

Thus we have

which gives the result (5.1).

Theorem 5.2. Let the infectious disease is spreading by the rulings F (x) = > 0, where, G(x) = e ^-x logE1 with the weight function P(x) = , and the parameters ρ ₀ > 0, bj ∈ ℝ, 0 < aj < 1∀ j = 1, 2,3,...,n; then, for ρ > ρ ₀ , the spreading area A of the disease in the domain (p0, p ) G R, is obtained by the formula ∆ = dx and found by

Here, the F ⁽ⁿ⁾ _D (.)being a Lauricella's multiple hypaergeometric function defined in (H. M. ^{Srivastava & Manocha, 1984}, Eqn.(4),p. 60).

Proof. Starting from (5.1), we have

Now suppose 0 <ρ₀<ρ_, x-ρ ₀ = (ρ₀-ρ₀)t ∀0 < ρ ₀ ≤ x ≤ ρ,t ≥0⁺ then

which on applying the technique of ^{Chandel, Agrawal and Kumar (Chandel, Agrawal, & Kumar, 1993}), and the formula of the ^{Exton (Exton, 1976}, Eqn. (2.3.6), p. 49) yields the result (5.2). □

Remark 5.1. In the Theorem 5.2,we have used the ruling,

which emphasize the following useful integral

Theorem 5.3. Let the infectious disease is spreading by the rulings F(x) = 0,where, G(x) = ; with the weight function P(x)= and the parameters bj ∈ ℝ, aj > 0, cj ≠ 0, - 1, -2,..., ∀j = 1,2,3,...,n, and │a ₁ +... + a _n │< 1. Then the spreading area ∆ of the disease in the domain (0,∞) ⊂ ℝ, is obtained by the formula ∆' = - and found by

Proof. In the similar manner of Theorem 5.2, we have

Then apply the Euler integral formula of ^{Exton (Exton, 1976}, Eqn. (2.4.2), p. 49)

to get the formula (5.4).The F ⁽ⁿ⁾ (.) being a Lauricella's multiple hypaergeometric function defined in (H. M. ^{Srivastava & Manocha, 1984}, Eqn. (1), p. 60).

Relations of logarithmic Mittag Leffler function into Riemann-Liouville fractional integral

From the Theorem 5.1, consider that

G(z) = e ^-z log E m = 2,3,4,..., and that the Riemann - Liouville fractional integral, given by (^{Diethelm, 2010}, p.13)

Therefore by proof of the Theorem 5.1, we have

Now, define that

and any function Φ (z) such that limz _→±∞ Φ(z)= 0.

Then for z > 0, make an appeal to the Theorem 5.1 and along with formulae (5.4) and (5.5), we obtain the relation of logarithmic function with the Riemann - Liouville fractional integral as

In another way, we write

Therefore, we get