Deformed exponential function for complex argument: writing roots of polynomial equations with complex coefficients

The generalization of the logarithm and exponential functions proposed by Tsallis in 1988 has brought contributions in several areas in the last decades. Recently, we introduced a deformed exponential function that may assume negative or even complex values, but with the constrain of admitting only real arguments. We also showed the possibility of writing the roots of polynomial equations with real coefficients by using our generalization. Here, we present an extension of our deformed exponential function that admits complex argument. We show that our new generalization can address all the roots of polynomial equations up to the third degree even with complex coefficients.

Keywords:
Complex systems; Generalized functions; Family of functions; Analyticity in the complex plane; Polynomial roots

Authorship

Paulo S. Adami ^**Correspondence email address: adami.paulo@ifsp.edu.br

Instituto Federal de Educação, Ciência e Tecnologia de São Paulo, Sertãozinho, SP, Brasil.Instituto Federal de Educação, Ciência e Tecnologia de São PauloBrasilSertãozinho, SP, BrasilInstituto Federal de Educação, Ciência e Tecnologia de São Paulo, Sertãozinho, SP, Brasil.

Universidade de São Paulo, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Ribeirão Preto, SP, Brasil.Universidade de São PauloBrasilRibeirão Preto, SP, BrasilUniversidade de São Paulo, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Ribeirão Preto, SP, Brasil.

https://orcid.org/0000-0003-3510-0215

Olavo H. Menin

https://orcid.org/0000-0002-2134-4166

Alexandre S. Martinez

Instituto Nacional de Ciência e Tecnologia de Sistemas Complexos, Maringá, PR, Brasil.Instituto Nacional de Ciência e Tecnologia de Sistemas ComplexosBrasilMaringá, PR, BrasilInstituto Nacional de Ciência e Tecnologia de Sistemas Complexos, Maringá, PR, Brasil.

https://orcid.org/0000-0002-4395-0511

*Correspondence email address: adami.paulo@ifsp.edu.br

SCIMAGO INSTITUTIONS RANKINGS

Figures | Formulas

Figures (2)
Formulas (12)

Thumbnail

Figure 1
Graphical representation of the generalized exponential function e_λ(x) for λ = −3. In the interval x < 1/3, the function has properties similar to those of the usual exponential function, such as being equal to 1 for x = 0, being positive and strictly increasing, but unlike the usual function,

\lim_{x \to 1 / 3 -} e_{λ} (x) = + \infty

, that is, it diverges at a finite value of the domain. For x > 1/3, the function e_λ(x) takes null values.

Thumbnail

Figure 2
Graphical representations for the deformed exponential function

{\tilde{e}}_{λ} (x)

, where the real part (Re) is represented by the solid line and the imaginary part (Im) by the dashed line. In (a), for λ = −2, unlike the Tsallis function, the deformed exponential admits nonzero values in the interval of x > 1/2, assuming complex values. For values of x < 1/2 the function only admits real values, behaves like the Tsallis function. It can be seen that the function diverges at a finite value of the domain, that is,

\lim_{x \to 1 / 2 -} {\tilde{e}}_{- 2} (x) = + \infty

. In (b), for λ = 1, the function behaves like a linear function, assuming negative values for x < −1. For λ = 1/5, Fig. (c) highlights the existence of an inflection point at x = −5 and the fact that

\lim_{x \to \infty -} {\tilde{e}}_{1 / 5} (x) = - \infty

. Such characteristics are not observed in the usual exponential function. In (d), for λ = 10 the loss of monotonicity can be observed.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Figure 1 Graphical representation of the generalized exponential function e_λ(x) for λ = −3. In the interval x < 1/3, the function has properties similar to those of the usual exponential function, such as being equal to 1 for x = 0, being positive and strictly increasing, but unlike the usual function, $\lim_{x \to 1 / 3 -} e_{λ} (x) = + \infty$ , that is, it diverges at a finite value of the domain. For x > 1/3, the function e_λ(x) takes null values.

Figure 2 Graphical representations for the deformed exponential function ${\tilde{e}}_{λ} (x)$ , where the real part (Re) is represented by the solid line and the imaginary part (Im) by the dashed line. In (a), for λ = −2, unlike the Tsallis function, the deformed exponential admits nonzero values in the interval of x > 1/2, assuming complex values. For values of x < 1/2 the function only admits real values, behaves like the Tsallis function. It can be seen that the function diverges at a finite value of the domain, that is, $\lim_{x \to 1 / 2 -} {\tilde{e}}_{- 2} (x) = + \infty$ . In (b), for λ = 1, the function behaves like a linear function, assuming negative values for x < −1. For λ = 1/5, Fig. (c) highlights the existence of an inflection point at x = −5 and the fact that $\lim_{x \to \infty -} {\tilde{e}}_{1 / 5} (x) = - \infty$ . Such characteristics are not observed in the usual exponential function. In (d), for λ = 10 the loss of monotonicity can be observed.

(1)

e_{λ} (x) = \lim_{λ^{'} \to λ} {(1 + λ^{'} x)}^{1 / λ^{'}} θ (1 + λ^{'} x),

(2)

{\tilde{e}}_{λ} (x) = \lim_{λ^{'} \to λ} {|1 + λ^{'} x|}^{1 / λ^{'}} e^{i π / λ^{'} [1 - θ (1 + λ^{'} x)]},

(3)

\begin{array}{l} {\tilde{E}}_{λ} (x + y i) & = & \lim_{λ^{'} \to λ} {[|1 + λ^{'} x| e^{i π [1 - θ (1 + λ^{'} x)]} + 1 + λ^{'} y i - 1]}^{1 / λ^{'}} \\ = & \lim_{λ^{'} \to λ} {[|1 + λ^{'} x| e^{i π [1 - θ (1 + λ^{'} x)]} + λ^{'} y i]}^{1 / λ^{'}}, \end{array}

(4)

\tilde{x} = 1 + (- \frac{b}{a} - 1) = 1 + γ = {\tilde{E}}_{1} (γ)

(5)

\begin{array}{l} {\tilde{E}}_{1} (γ) & = & |1 + [Re (z_{1}) - 1]| e^{i π {1 - θ [1 + Re (z_{1}) - 1]}} + Im (z_{1}) i \\ = & |Re (z_{1})| e^{i π {1 - θ [Re (z_{1})]}} + Im (z_{1}) i \\ = & Re (z_{1}) + Im (z_{1}) i = \tilde{x} . \end{array}

(6)

\begin{array}{l} {\tilde{x}}_{\pm} & = & - \frac{b}{2 a} (1 \pm \sqrt{1 - \frac{4 a c}{b^{2}}}) \\ = & α (1 \pm \sqrt{1 + 2 β}) = α [1 \pm {(1 + 2 β)}^{1 / 2}] \\ = & α [1 + Re (z_{\pm}) + Im (z_{\pm}) i], \end{array}

(7)

\begin{array}{l} {\tilde{x}}_{\pm} & = & α [1 \pm {(1 + 2 β)}^{1 / 2}] = α [1 \pm {\tilde{E}}_{2} (β)] \\ = & α {{\tilde{E}}_{1} [\pm {\tilde{E}}_{2} (β)]} . \end{array}

(8)

\begin{array}{l} {\tilde{E}}_{2} (β) & = & {[1 + 2 Re (β) + 2 Im (β) i]}^{1 / 2} \\ = & {(1 + 2 β)}^{1 / 2} = z_{\pm} . \end{array}

(9)

{\tilde{E}}_{1} [{\tilde{E}}_{2} (β)] = 1 + Re (z_{\pm}) + Im (z_{\pm}) i .

(10)

α {{\tilde{E}}_{1} [{\tilde{E}}_{2} (β)]} = α [1 + Re (z_{\pm}) + Im (z_{\pm}) i] = {\tilde{x}}_{\pm} .

(11)

Δ_{\pm} = {\tilde{E}}_{3} \{\frac{- \frac{q}{2} {\tilde{E}}_{1} [\pm {\tilde{E}}_{2} (ϕ)] - 1}{3}\},

(12)

\begin{array}{l} {\tilde{x}}_{1} & = & - γ + Δ_{+} + Δ_{-}, \\ {\tilde{x}}_{2} & = & - γ + χ_{+} Δ_{+} + χ_{-} Δ_{-}, \\ {\tilde{x}}_{3} & = & - γ + χ_{-} Δ_{+} + χ_{+} Δ_{-} . \end{array}

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Deformed exponential function for complex argument: writing roots of polynomial equations with complex coefficients

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[1] *Correspondence email address: adami.paulo@ifsp.edu.br