[Paper Review] Complex Multiplicative Calculus

Agamirza E. Bashirov, Mustafa Riza|arXiv (Cornell University)|Mar 8, 2011

Mathematical and Theoretical Analysis8 references4 citations

TL;DR

This paper extends multiplicative calculus to complex-valued functions of a complex variable, introducing complex multiplicative derivatives and integrals. Unlike ordinary complex calculus, it resolves real-case limitations—such as handling negative and zero values—by leveraging the periodicity of the complex exponential, and shows that the Cauchy integral formula vanishes due to $ e^{2/pi i} = e^0 = 1 $, despite $ 2\pi i \neq 0 $, revealing a fundamentally distinct calculus framework.

ABSTRACT

In the present paper we extend the concepts of multiplicative de- rivative and integral to complex-valued functions of complex variable. Some drawbacks, arising with these concepts in the real case, are explained satis- factorily. Properties of complex multiplicative derivatives and integrals are studied. In particular, the fundamental theorem of complex multiplicative calculus, relating these concepts, is proved. It is shown that complex multi- plicative calculus is not just another realization of the ordinary calculus. In particular, the Cauchy formula of complex calculus disappears in multiplicative complex calculus.

Motivation & Objective

• To extend multiplicative calculus to complex-valued functions of a complex variable, overcoming limitations in the real case.
• To resolve unresolved issues in real multiplicative calculus, such as applicability to negative functions and the role of zero.
• To define and study complex multiplicative derivatives and integrals, establishing their geometric and analytical properties.
• To establish a fundamental theorem linking complex multiplicative derivatives and integrals.
• To demonstrate that complex multiplicative calculus is not equivalent to ordinary complex calculus, particularly by showing the disappearance of the Cauchy integral formula.

Proposed method

• Define the complex multiplicative derivative as $ f^*(z) = \lim_{h \to 0} \left( \frac{f(z+h)}{f(z)} \right)^{1/h} $, which simplifies to $ f^*(z) = e^{\mathrm{Log}(f)^\prime(z)} $ for nowhere-vanishing holomorphic functions.
• Define the complex multiplicative integral as a product over partitions: $ \int_C f(z)^{dz} = \lim_{\mathcal{P}} \prod_{k=1}^m f(z_k)^{z_k - z_{k-1}} $, with multivaluedness arising from logarithmic branches.
• Use the principal branch of the complex logarithm $ \mathrm{Log}\,z = \ln|z| + i\mathrm{Arg}\,z $ to relate multiplicative calculus to ordinary calculus via exponentiation.
• Prove the fundamental theorem of complex multiplicative calculus: $ \int_C f^*(z)^{dz} = e^{2\pi i n (z(b) - z(a))} \frac{f(z(b))}{f(z(a))} $, where $ n \in \mathbb{Z} $.
• Demonstrate that the Cauchy integral $ \oint_{|z|=1} \frac{dz}{z} = 2\pi i $ becomes trivial in multiplicative calculus: $ \oint_{|z|=1} (e^{1/z})^{dz} = 1 $, due to $ e^{2\pi i} = 1 $.
• Use examples such as $ f(z) = e^{cz} $ and $ f(z) = e^{c e^z} $ to verify the behavior of multiplicative derivatives and integrals.

Experimental results

Research questions

• RQ1 How can multiplicative calculus be consistently extended to complex-valued functions, including those with negative or zero values?
• RQ2 Why do real multiplicative calculus issues—such as the positivity of higher-order derivatives of negative functions—disappear in the complex setting?
• RQ3 What is the relationship between complex multiplicative derivatives and integrals, and how does it differ from the fundamental theorem of ordinary calculus?
• RQ4 Why does the classical Cauchy integral formula $ \oint \frac{dz}{z} = 2\pi i $ fail to hold in complex multiplicative calculus?
• RQ5 Can the multivalued nature of complex multiplicative integrals be systematically characterized and linked to topological properties of the domain?

Key findings

• The complex multiplicative derivative is single-valued and defined via $ f^*(z) = e^{\mathrm{Log}(f)^\prime(z)} $, enabling application to nowhere-vanishing holomorphic functions on open connected sets.
• The complex multiplicative integral is multivalued and given by $ \int_C f^*(z)^{dz} = e^{2\pi i n (z(b) - z(a))} \frac{f(z(b))}{f(z(a))} $, with $ n \in \mathbb{Z} $, reflecting monodromy from logarithmic branches.
• For a closed curve $ C $, the multiplicative integral simplifies to $ \oint_C f^*(z)^{dz} = 1 $, independent of the path, showing it becomes single-valued on loops.
• The classical Cauchy formula $ \oint \frac{dz}{z} = 2\pi i $ is absent in multiplicative calculus because $ e^{2\pi i} = e^0 = 1 $, despite $ 2\pi i \neq 0 $, demonstrating a fundamental structural difference.
• The function $ f(z) = e^{cz} $ has multiplicative derivative $ f^*(z) = e^c $, and its multiplicative integral over a curve $ C $ is $ e^{(z(b)-z(a))(c + 2\pi n i)} $, confirming the general formula.
• The example $ f(z) = e^{c e^z} $ has $ f^*(z) = f(z) $, and its multiplicative integral is $ e^{2\pi i n (z(b)-z(a))} e^{c(e^{z(b)} - e^{z(a)})} $, verifying the fundamental theorem in a non-trivial case.