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C being any closed curve in the angular region surrounding C1 . 1 f z dz 1 f z dz 1 Hint: Write f a h and expand in two di erent 2i C2 z a h 2i C1 z a h z a h ! ways. z which converges for 1 < jzj < 2 and 16.93. Find a Laurent series expansion for the function f z z 1 z 2 diverges elsewhere. ! z 1 2 1 1 Hint: Write : z 1 z 2 z 1 z 2 z 1 1=z 1 z=2
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FUNCTIONS OF A COMPLEX VARIABLE 1 1 1 1 1 z z2 z3 1 8 2 4 z5 z4 z3 z2 z
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Ans: 1 16.94. Let
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e st F t dt f s where f s is a given rational function with numerator of degree less than that of the If C is a simple closed curve enclosing all the poles of f s , we can show that 1 F t ezt f z dz sum of residues of ezt f z at its poles 2i C (a) s ; 1 b 1 ; 2s 5 c s2 1 ; s s 1 2 d 1 and s2 1 2
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[Note that f s is the Laplace transform of F t , and F t is the inverse Laplace transform of f s (see 12). Extensions to other functions f x are possible.] Ans. (a) cos t; b 1 e t sin 2t; c 1 5 te2t 3 e2t ; d 1 sin t t cos t 2 4 2 4 2
Abel, integral test of, 334 summability, 305 theorems of, 282, 293 Absolute convergence: of integrals, 309, 312, 319 321 of series, 268, 283, 300 theorems on, 269, 283 Absolute maximum or minimum, 42 (See also Maxima and minima) Absolute value, 3 of complex numbers, 6 Acceleration, 74, 158 centripetal, 175 in cylindrical and spherical coordinates, 181 normal and tangential components of, 177 Accumulation, point of, 5, 117 (See also Limit points) Addition, 1 associative law of, 2, 8 commutative law of, 2 of complex numbers, 7, 13 of vectors, 151, 152, 163 Aerodynamics, 402 Aleph-null, 5 Algebra: fundamental theorem of, 43 of complex numbers, 6, 7, 13 15 of vectors, 151, 152, 163 171 Algebraic functions, 43 Algebraic numbers, 6, 13 countability of, 13 Alternating series, 267, 268, 282 convergence test for, 267, 268 error in computations using, 268, 282 Amplitude, 7 Analytic continuation, of gamma function, 376 Analytic functions, 393
Analytic part, of a Laurent series, 395 Anti-derivatives, 94 Approximations (see Numerical methods) by partial sums of Fourier series, 352 by use of di erentials, 78, 79, 130, 131 least square, 201 to irrational numbers, 9 using Newton s method, 74 using Taylor s theorem, 274 275 Archimedes, 90 Arc length, 99, 109 element, 157, 161, 174 Area, 100, 109 expressed as a line integral, 242 of an ellipse, 205 of a parallelogram, 155, 168 Argand diagram, 7 Argument, 7 Arithmetic mean, 10 Associative law, 2, 4 for vectors, 152, 155 Asymptotic series or expansions: for gamma function, 286, 292 Axiomatic foundations: of complex numbers, 6, 7 of real number, 4 of vector analysis, 155 Axis, real, 2 x, y and z, 121 Base, of logarithms, 4 Bases, orthonormal, 152 Bernoulli, Daniel, 336 Bernoulli numbers, 304 Bernoulli s inequality, 16 Bessel functions, 276 Bessel di erential equation, 276
Beta functions, 375, 378, 379, 382, 384 relation, to gamma functions, 379 Bilinear transformation, 422 (See also Fractional linear transformation) Binary scale, 16, 21 system, 2 Binomial coe cients, 21 series, 275 theorem, 21 Bolzano-Weierstrass theorem, 6, 12, 19, 117 Boundary conditions, 339 Boundary point, 117 Boundary-value problems: and Fourier integrals, 371 and Fourier series, 339, 356, 357 in heat conduction, 356, 357 in vibration of strings, 361 separation of variables method for solving, 356 Bounded functions, 40, 41 sequences, 24, 30 32, 36 sets, 6 Bounds, lower and upper, 6, 12, 13 Box product, 155 Branches of a function, 41 Branch line, 417 Branch point, 396, 417 Calculus, fundamental theorem of integral, 94, 95, 104 Cardinal number of the continuum, 5 Cardioid, 114 Catenary, 113 Cauchy principal value, 310, 321 Cauchy-Riemann equations, 393, 400 403 derivation of, 401
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