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.
About the Authors’
Mr. A. Durai Ganesh is an Assistant Professor in the Department of Mathematics at PET Engineering College, Tamil Nadu. He is committed to excellence in mathematics education, inspiring students through innovative teaching, analytical thinking, and problem-solving approaches while actively contributing to academic development and scholarly pursuits in the field of mathematics.
Dr. Brinda Halambi is currently working as an Associate Professor in the Department of Mathematics, School of Applied Sciences at REVA University, Bangalore, INDIA. She has obtained a Ph.D. in Mathematics from REVA University Bangalore, INDIA. Her main research interests include fluid mechanics, dynamics, tribology, and heat and mass transfer.
Dr. Archana Shukla, is presently working as Head, Department of Mathematics, Nehru Gram Bharati University, Prayagraj. She received Ph.D degree in the field of Fluid Mechanics. She has 8 years teaching experience as well as administrative experience as a proctorial board. She has presented and published a number of research papers in reputed Journals and one patent. Two students awarded Ph.D degree under her supervision and now six is going on. She was honoured by the Silver Jubilee award from IAPS and recently, She has been honoured by the best teacher award from Nehru Gram Bharati University in 2022, Prayagraj, Uttar Pradesh.
Dr. S. Ulagammal is an Assistant Professor in the Department of Mathematics at Thamirabharani Engineering College (Autonomous), Tirunelveli, Tamil Nadu, India. She is dedicated to mathematics education, research, and academic excellence, with a strong focus on innovative teaching, analytical problem-solving, and fostering students’ quantitative and critical-thinking skills in higher education.
Palanivelu Saranya is an Assistant Professor in the Department of Mathematics at Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology. Her academic interests include engineering mathematics and applied mathematical sciences. She is dedicated to teaching, research, and nurturing students’ mathematical aptitude and critical-thinking abilities.
