Chap 4 Complex Algebra
For application to Laplace Transform Complex Number
Argand Diagram r x y
Complex Variables Continuous Function Cplxdemo.m
Single Value Function Many Values Function
Derivatives of Complex Variables 1 0
10 Cauchy Riemann Conditions
Analytic Functions It has single value in the region R It has a unique finite value It has a unique finite derivative at z 0, satisfies the Cauchy Riemann Conditions
Example
Cauchy Riemann Conditions
At Origin Keep y constant One_OVER_Z.m
Singularities Poles or unessential Essential Branch points
Poles or unessential Singularities Second order Poles Pole at a Pole order p at zero Pole order q at a
Essential Singularities E_1_z.m
Branch Points Many Value Function Single
4.13 INTEGRATION OF FUNCTION OF COMPLEX VARIABLES
Cauchy’s Theorem ถ้ามีฟังก์ชั่นใดที่เป็น Analytic ภายในหรือบน closed contour, integration รอบ contour จะได้ศูนย์ Stake’s theorem Cauchy – Riemann conditions integral ทางด้านขวามือจะเป็นศูนย์
ตามเส้นทาง AB หรือ รอบเส้นทาง ACDB path AB
curve ACDB 1. ตาม AC
2. เส้นโค้ง CDB ซึ่งมี constant radius 10 ผลรวมของ Integral
Example 2 Evaluate around a circle with its center at the origin. Although the function is not analytic function
Example 3 Evaluate around a circle with its center at the origin. This result is one of the fundamentals of contour integration
Cauchy’s Integral formula f ( a ) =constant at
The theory of Residue Pole at origin Laurent expansion
Example 1 Evaluate if Around a circle center at the origin Function is analytic if There is a pole order 3 at z = a
Evaluation without Laurent expansion Many poles : independently evaluate
Example 2 Evaluate the residues of Poles at 3,-4 Sum of Residues = 1
If the denominator does not factorize L’Hopital’s rule
Example 4 evaluate Around circle and Pole at z = 0
Multiple Poles Dividing throughout by
Example 5 Evaluate