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