Introduction to Complex Variables and Applications |
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Page 76
... integrand is a continuous function of t . Likewise , the last integral in equation ( 2 ) can be written as a definite integral of a continuous function of t . The line integral Sf ( z ) dz therefore exists when the path C has the ...
... integrand is a continuous function of t . Likewise , the last integral in equation ( 2 ) can be written as a definite integral of a continuous function of t . The line integral Sf ( z ) dz therefore exists when the path C has the ...
Page 89
... integrand has the values prescribed above in the upper half plane . Hence S11 z dz = fr2 + 3i0 / 2 | 1 = 1 , = { ( 1 - - eзiπ / 2 ) = ( 1 + i ) . 1 12 : The integral ( 4 ) over every path below the x axis has another value . The integrand ...
... integrand has the values prescribed above in the upper half plane . Hence S11 z dz = fr2 + 3i0 / 2 | 1 = 1 , = { ( 1 - - eзiπ / 2 ) = ( 1 + i ) . 1 12 : The integral ( 4 ) over every path below the x axis has another value . The integrand ...
Page 119
... integrand lie inside C. To find the 0 , we may write - - - 1 Ki Z ( ( 2 ) ( 2 = 2 ) ( 1 + 2 + 22 + • • • ) 2-116 3 3z 322 - when 0 << 1. Hence K1 | - = 2 . To find the residue K2 at z = 1 , we may use the Taylor series 1 = 1 ...
... integrand lie inside C. To find the 0 , we may write - - - 1 Ki Z ( ( 2 ) ( 2 = 2 ) ( 1 + 2 + 22 + • • • ) 2-116 3 3z 322 - when 0 << 1. Hence K1 | - = 2 . To find the residue K2 at z = 1 , we may use the Taylor series 1 = 1 ...
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absolute value According to equation analytic continuation analytic function angle Appendix approaches zero arctan boundary branch cut C₁ C₂ Cauchy-Goursat theorem Cauchy-Riemann conditions Cauchy's integral formula closed curve coefficients complex number complex potential complex variable conformal mapping conjugate continuous function corresponding cosh defined definition denote example EXERCISES exists Find finite number flow fluid follows formula function f(z function is analytic half plane harmonic function Hence inequality infinite integrand interior inverse Laurent series limit line integral linear fractional transformation Maclaurin series maps MICHIGAN neighborhood partial derivatives path pole polygon positive number power series quadrant R₁ real axis real numbers represents residue Riemann surface satisfied single-valued and analytic single-valued function sinh steady temperatures strip tends to infinity unit circle upper half vanishes vector velocity w₁ write xy plane z₁ πί ди дх მყ