Partial Differential Equations with Probability and Complex Functions

Partial Differential Equations with Probability and Complex Functions presents a systematic and comprehensive treatment of important mathematical concepts that form an essential part of undergraduate and postgraduate mathematics and engineering curricula. The book brings together Z-Transforms and Difference Equations, Partial Differential Equations, Probability and Random Variables, and Complex Integration in a structured and progressive manner. The opening chapter develops the theory and applications of Z-Transforms, including linearity, differentiation in the Z-domain, shifting theorems, unit impulse and unit step sequences, initial and final value theorems, and inverse Z-Transforms. The chapter on Partial Differential Equations provides detailed coverage of formation of partial differential equations, elimination of arbitrary constants and functions, singular integrals, standard first-order equations, Lagrange’s linear equation, and higher-order linear partial differential equations with constant coefficients. The treatment emphasizes systematic solution procedures and illustrative problems to strengthen conceptual understanding and mathematical problem-solving skills. The Probability and Random Variables chapter introduces fundamental probability concepts, axioms of probability, conditional probability, and Bayes’ theorem, supported by suitable examples and applications. The final chapter develops the foundations of Complex Integration through line integrals, Cauchy’s Integral Theorem, Cauchy’s Integral Formula, Taylor’s and Laurent’s series, singularities, residues, and residue theorems. Throughout the book, definitions, fundamental results, derivations, worked examples, and problem-solving approaches are integrated to facilitate effective learning. The presentation progresses from fundamental concepts to more advanced techniques, making the material accessible while maintaining mathematical rigor. The book is designed to support classroom teaching, university examination preparation, independent study, and further exploration of applied mathematical methods. Its integrated coverage makes it a valuable academic resource for students, teachers, researchers, and professionals working with mathematical analysis and its applications in science, engineering, and technology.