Complex Analysis Presentation

1. Uttara university Welcome To Our Presentation

2. “Complex Analysis” Presentation on

3. Presenting By Name: Md. Esa Mia ID: 2231151032 Name: Md. Sobuj Ali ID: 2233151040 Department of Mathematics Uttara University PresentingTo Professor Dr. Shahansha Khan Professor and Chairman Uttara University

4. OverView • Introduction • Complex Number • Geometric Representation • Complex Function • Function of complex variable • Differentiability in complex analysis • Cauchy's-Riemann Equation • Analytic Function • Harmonic Function • Contour Integration & Cauchy's theorem • Zeros and Poles • Singularities • Cauchy's Integral formula • Cauchy's Residue theorem • Taylor and Laurent Series • Limits and Continuity • Application of Complex Analysis • Summary • Conclusion

5. Introduction Complex Analysis is a branch of mathematics that studies functions of a complex variable. Unlike real numbers, complex numbers have both real and imaginary parts, which allows us to explore mathematical concepts in a richer and more powerful way. In this subject, we learn about analytic functions, limits, differentiation, integration, and various theorems that simplify complicated calculations. Complex Analysis plays an important role in physics, engineering, fluid flow, signal processing, and many modern scientific applications.

6. What Are Complex Numbers? • Complex numbers extend the idea of real numbers by including an imaginary component. A complex number is written as: • z=x+iy • Where, • x= Real part • y= Imaginary Part • i is the imaginary unit with the property =-1 • The set of all Complex numbers is denoted by ℂ. • Why do we need complex number? • They allow us to solve equations like which have no real solution.

7. Y Properties Equality of complex number : Let two complex numbers + Both are equal when y θ X x Modulas of complex number : Let z=x+iy be the complex number. Then |z|= Argument of complex number : If z=x+iy be the any complex number then arg z= ∴Principal Argument =arg z= Then general argument is, 2nπ+ principal Argument Addition of complex number : Let + )=(

8. Geometric Representation • Complex numbers can be represented on the Argand Plane (also called the complex plane). • Horizontal axis = Real axis • Vertical axis = Imaginary axis • Every complex number corresponds to a unique point in this plane. • We can also write complex numbers in polar form: • Z=r • Where • r= |z|= Magnitude • =arg(z) = argument (angle)

9. Complex Function A Function f(z)=U(x,y) + iV(x,y), where U and V are real valued. Example: f(z)= f(z)=

10. Functions of a Complex Variable A complex function is a rule that assigns each complex number z to another complex number w; f : C → C ;It is denoted w=f(z) Y Y • Example: • f(z)= • f(z)= • f(z)= • f(z)=Sin(z) (U,V) (x,y) z=x+iy z=u+iv W-Plane X X Z-Plane O O These functions often behave more smoothly than real functions, which is a key reason complex analysis is so powerful.

11. Differentiability in Complex Analysis A function f(z) is differentiale at a point if : exists and is independent of the direction from which h approaches 0. Unlike real analysis, Complex differentiability is a very strong condition.

12. Cauchy–Riemann Equations If f(z) = U(x,y) + iV(x,y) is differentiable at a point, then it must satisfy the Cauchy-Riemann (CR) equation: = & = - These equations connect the real and imaginary parts of the function. If the CR equations hold and ‘U,V’ have continuous partial derivatives, then f is analytic.

13. Analytic (Holomorphic) Function A single valued complex function f(x) is said to be analytic in the region if existin the region. • Analytic function have many beautiful properties: • They are infinitely differentiable • They can be expressed as power series • They have no sudden jumps or corners. • Example of analytic function: • Polynomials • Exponential function • Trigonometric functions Sin(z),Cos(z) • Rational functions (except at poles).

14. Hermonic Function Let, f(z) be the single valued function f(z) = U(x,y) + V(x,y) Where, U(x,y) = Real part V(x,y) = Imaginary part Then function is harmonic when it satisfy the laplace equation. , ∴ + =0, + + • Hermonic Conjugate: • U is given,V is harmonic conjugate • V is given, U is harmonic conjugate

15. Contour Integration • What is Contour? • Contour is a directed curve in the complex plane. • It may be open or closed. • A simple closed contour does not intersect itself. • Example: • Circle • Rectangle • Any smooth closed curve Contour Integration: Contour integration is the integration of a complex function along a specified path (contour) in the complex plane. Then:

16. Cauchy’s Theorem Cauchy’s Theorem: If f(z) is analytic inside and on a simple closed curve c, then = 0 C f(z) • Conditions for Cauchy’s Theorem • Cauchy’s Theorem is valid if: • The function is analytic • No singularities inside the contour • The contour is closed • The region is simply connected R

17. Example of Cauchy’s Theorem • Evaluate: • Solution: • f(z) = • Polynomial function are analytic everywhere • By Cauchy’s Theorem: • = 0

18. Zeros of Complex Function • A point is called a zero of a complex function f(z) if • f(z) = 0 • Types of Zeros • Simple Zero: Zero of order 1 • Multiple Zero: Zero of order greater than 1 • Example: f(z) =

19. Pole of Complex Function A point is called a pole of a function f(z) if the value of the function becomes infinite as z →. Types of Poles Simple pole (order 1) Pole of order of n Example: f(z) =

20. Singularities A singularity is a point at which a complex function is not analytic. If a function f(z) is not analytic at z = , then is called a singular point.

21. Types of Singularity Essential Singularity Removable Singularity Pole If the function becomes infinite at the point. Example: f(z) = If the function shows unpredictable behavior near the point. Example: f(z) = If a singularity can be removed by redefining the function at that point. Example: f(z) =

22. Difference between Zeros and Poles Zeros Poles Funtion value is zero Function remains finite Example: f(z) = z at z = 0 Function value is infinity Function becomes unbounded Example: f(z) = at z = 0

23. Cauchy’s Integral Formula If f(z) be the analytic function on a closed curve and on its boundary and a be the any points in the closed curve f(a) = dz Even derivatives: dz So analytic functions are infinitely differentiable!

24. Cauchy’s Residue Theorem • Residue theorem is one of the strongest tools in evaluate complex integrals: • = 2πi ∑Residues inside c. • This theorem transforms difficult real integrals into simple complex calculations. • Example: dz • Solution: • Singularity at z = 2 • z = 2 lies inside |z| = 3 • It is simple pole Residue at z = 2: = 1 Result dz = 2πi ×

25. Why two theorems? (Cauchy’s Integral formula VS Residue theorem) • When we have a simple pole and the numerator is analytic → Cauchy’s Integral Formula. • But for higher-order poles, or complicated numerators → Integral Formula becomes messy (derivatives everywhere). Residue Theorem handles everything in one line. • So Integral Formula = beautiful shortcut for the easy case. Residue Theorem = the master weapon for all cases. • Basically, one theorem could do everything… but mathematicians love style, so we got two!

26. Taylor and Laurent series Analytic function can be expressed as power series: Taylor series: Valid in regions where the function is analytic. f(z) = • Laurent Series: Used around singularities (points where f is not analytic). • f(z)= • Laurent series helps classify singularities: • Removable • Pole • Essential

27. Limits and Continuity The idea of limit must be the same regardless of the direction from which z approaches a point. Example of a limit that does not esist: Since different directions gives different values, the limit fails to exist. A function is continuous if the limit as z approaches any point equals the function value at that point.

28. Applications of Complex Analysis Complex analysis is used in many fields: • Physics: • Electromagnetic fields • Quantum mechanics • Fluid flow and potential theory • Engineering: • Signal processing • Control theory • Electrical circuits • Mathematics: • Evaluate real integrals • Solve differential equations • Conformal mapping • Technology: • Image processing • Computer graphics • Data compression

29. Summary Complex numbers extend real numbers and are represented in the complex plane Complex functions behave more smoothly than real functions Cauchy–Riemann equations determine differentiability Analytic functions are infinitely differentiable Cauchy’s theorems form the foundation of complex integration Complex analysis has huge real-world applications

30. Conclusion Complex Analysis is one of the most elegant branches of mathematics. It beautifully connects geometry, calculus, and algebra. The power of analytic functions and the simplicity of complex integrals make this subject both useful and fascinating.

31. Reference • Brown, J.W., & Churchill, R.V. Complex Variables and Applications. McGraw-Hill. • Zill, D.G., & Shanahan, P.D. A First Course in Complex Analysis with Applications. Jones & Bartlett. • Stein, E.M., & Shakarchi, R. Complex Analysis. Princeton University Press. • Schaum's Outline Series, Complex Variables (Schaum’s Outline). • Professor’s Lecture Notes (Class Handouts & PDF Materials). • MIT OpenCourseWare – Complex Variables Lecture Notes. • Wolfram MathWorld – Complex Analysis Topics. • ChatGPT (OpenAI) – Concept Explanation & Draft Assistance. • Google Gemini – Additional Explanation Support. • xAIGrok – Cross-Checking & Supplementary Insights.