27 0 obj /Count 6 /Trapped /False << >> << >> 15 0 obj /Kids [154 0 R 155 0 R 156 0 R 157 0 R 158 0 R 159 0 R] endobj /Parent 7 0 R Solution… 692.5 323.4 569.4 323.4 569.4 323.4 323.4 569.4 631 507.9 631 507.9 354.2 569.4 631 /F 2 >> %���� 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 << Here is a set of practice problems to accompany the Integration by Parts section of the Applications of Integrals chapter of the notes for Paul Dawkins Calculus II course at Lamar University. /OpenAction 5 0 R A tutorial on how to find the conjugate of a complex number and add, subtract, multiply, divide complex numbers supported by online calculators. 1 0 obj The calculus page problems list. /FontDescriptor 15 0 R endobj /Name/F3 Example Find an upper bound for Z Γ ez/(z2 + 1) dz , where Γ is the circle |z| = 2 traversed once in the counterclockwise direction. chapter 03: de moivre’s theorem. 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 /PageMode /UseOutlines /Type/Encoding << >> Practising these problems will encourage students to grasp the concept better. /Count 6 639.7 565.6 517.7 444.4 405.9 437.5 496.5 469.4 353.9 576.2 583.3 602.5 494 437.5 truth! endobj 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /LastChar 196 >> COMPLEX ANALYSIS: SOLUTIONS 5 5 and res z2 z4 + 5z2 + 6;i p 3 = (i p 3)2 2i p 3 = i p 3 2: Now, Consider the semicircular contour R, which starts at R, traces a semicircle in the upper half plane to Rand then travels back to Ralong the real axis. /Title (Foreword) It is worth pointing out that integration by substitution is something of an art - and your skill at doing it will improve with practice. /Count 37 endobj /LastChar 196 1074.4 936.9 671.5 778.4 462.3 462.3 462.3 1138.9 1138.9 478.2 619.7 502.4 510.5 >> >> /Type /Pages 506.3 632 959.9 783.7 1089.4 904.9 868.9 727.3 899.7 860.6 701.5 674.8 778.2 674.6 /FontDescriptor 26 0 R If values of three variables are known, then the others can be calculated using the equations. After a brief review of complex numbers as points in the complex plane, we will ﬂrst discuss analyticity and give plenty of examples of analytic functions. /Type /Pages 530.4 539.2 431.6 675.4 571.4 826.4 647.8 579.4 545.8 398.6 442 730.1 585.3 339.3 /Kids [57 0 R 58 0 R 59 0 R 60 0 R 61 0 R 62 0 R] >> /Kids [75 0 R 76 0 R 77 0 R 78 0 R 79 0 R 80 0 R] << << /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 >> /Kids [105 0 R 106 0 R 107 0 R 108 0 R 109 0 R 110 0 R] /D (Item.259) /Count 6 /Outlines 3 0 R 32 0 obj 18 0 obj 160/space/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 173/Omega/alpha/beta/gamma/delta/epsilon1/zeta/eta/theta/iota/kappa/lambda/mu/nu/xi/pi/rho/sigma/tau/upsilon/phi/chi/psi/tie] >> 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 /Type /Pages /Parent 8 0 R Of course, no project such as this can be free from errors and incompleteness. /Limits [(Doc-Start) (subsection.4.3.1)] << Now that complex numbers are defined, we can complete our study of solutions to quadratic equations. I = Z b a f(x)dx … Z b a fn(x)dx where fn(x) = a0 +a1x+a2x2 +:::+anxn. /Parent 8 0 R << /Resources 38 0 R Fall 02-03 midterm with answers. >> /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 161/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus (1.17) On the other hand, the diﬀerential form dz/z is closed but not exact in the punctured plane. 31 0 obj /Count 6 339.3 892.9 585.3 892.9 585.3 610.1 859.1 863.2 819.4 934.1 838.7 724.5 889.4 935.6 << << stream Next we seek an upper bound M for the function ez/(z2 + 1) when |z| = 2. >> /Subtype/Type1 465 322.5 384 636.5 500 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Parent 7 0 R >> /Parent 9 0 R /BaseFont/DIPVPJ+CMSY10 /D [13 0 R /Fit] /Next 32 0 R /Keywords () LECTURE 6: COMPLEX INTEGRATION 3 have R C dz zn = 0 where C is given by a circle of radius r around 0 (which we already know from the fundamental integral). /Encoding 7 0 R endobj 30 0 obj 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 >> Solution The path of integration has length L = 4π. Spring 03 midterm with answers. 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 /Subject () endobj /Kids [93 0 R 94 0 R 95 0 R 96 0 R 97 0 R 98 0 R] << It also connects widely with other branches of mathematics. /BaseFont/QCGQLN+CMMI10 /Filter /FlateDecode /Producer (pdfTeX-1.40.16) 13 0 obj /Count 6 << >> 7.2.1 Worked out examples /Kids [39 0 R 13 0 R 40 0 R 41 0 R 42 0 R 43 0 R 44 0 R] /Kids [45 0 R 46 0 R 47 0 R 48 0 R 49 0 R 50 0 R] 22 0 obj 797.6 844.5 935.6 886.3 677.6 769.8 716.9 0 0 880 742.7 647.8 600.1 519.2 476.1 519.8 /Parent 8 0 R << /Kids [7 0 R 8 0 R 9 0 R] endobj /Differences[0/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus/circlemultiply/circledivide/circledot/circlecopyrt/openbullet/bullet/equivasymptotic/equivalence/reflexsubset/reflexsuperset/lessequal/greaterequal/precedesequal/followsequal/similar/approxequal/propersubset/propersuperset/lessmuch/greatermuch/precedes/follows/arrowleft/arrowright/arrowup/arrowdown/arrowboth/arrownortheast/arrowsoutheast/similarequal/arrowdblleft/arrowdblright/arrowdblup/arrowdbldown/arrowdblboth/arrownorthwest/arrowsouthwest/proportional/prime/infinity/element/owner/triangle/triangleinv/negationslash/mapsto/universal/existential/logicalnot/emptyset/Rfractur/Ifractur/latticetop/perpendicular/aleph/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/union/intersection/unionmulti/logicaland/logicalor/turnstileleft/turnstileright/floorleft/floorright/ceilingleft/ceilingright/braceleft/braceright/angbracketleft/angbracketright/bar/bardbl/arrowbothv/arrowdblbothv/backslash/wreathproduct/radical/coproduct/nabla/integral/unionsq/intersectionsq/subsetsqequal/supersetsqequal/section/dagger/daggerdbl/paragraph/club/diamond/heart/spade/arrowleft /Title (1 Complex Numbers) /Parent 3 0 R << /Subtype/Type1 /LastChar 196 >> >> /Count 5 24 0 obj 20 0 obj << /Type /Pages /BaseFont/VYRNZU+CMMI7 /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 6 Integration: to solve complex environmental problems unintended negative consequences, or create new environmental or socio-economic problems12. 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 /Title (4 Series) /Prev 145 0 R 33 0 obj 16 0 obj 10 0 obj 6.2.1Worked out Examples . /ModDate (D:20161215200015+10'00') endobj 6.2.2 Tutorial Problems . << 9. << >> /Limits [(Doc-Start) (Item.56)] 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 49 integration problems with answers. 588.6 544.1 422.8 668.8 677.6 694.6 572.8 519.8 668 592.7 662 526.8 632.9 686.9 713.8 endobj >> >> >> endobj endobj Quadratic Equations with Complex Solutions. 12 0 obj /Prev 34 0 R /FirstChar 33 600.2 600.2 507.9 569.4 1138.9 569.4 569.4 569.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Step 1: Add one to the exponent Step 2: Divide by the same. 500 555.6 527.8 391.7 394.4 388.9 555.6 527.8 722.2 527.8 527.8 444.4 500 1000 500 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] This is done with a help of numerous examples and problems with detailed solutions. /Count 6 /Name/F4 13 0 obj /A 140 0 R The pages that follow contain “unofﬁcial” solutions to problems appearing on the comprehensive exams in analysis given by the Mathematics Department at the University of Hawaii over the period from 1991 to 2007. /Type /Pages << endobj >> /Encoding 21 0 R contents: complex variables . /Type /Pages >> /F 2 /Parent 7 0 R 3 0 obj /Parent 9 0 R /BaseFont/GDTASL+CMR10 /Subtype/Type1 /A 33 0 R The majority of problems are provided with answers, detailed procedures and hints (sometimes incomplete solutions). Examples and questions with detailed solutions on using De Moivre's theorem to find powers and roots of complex numbers. /Parent 8 0 R /Widths[719.7 539.7 689.9 950 592.7 439.2 751.4 1138.9 1138.9 1138.9 1138.9 339.3 >> In fact, to a large extent complex analysis is the study of analytic functions. 36 0 obj endobj endobj endobj /Count 6 /MediaBox [0 0 595.276 841.89] /Limits [(Item.57) (subsection.4.3.1)] /Parent 3 0 R /Kids [63 0 R 64 0 R 65 0 R 66 0 R 67 0 R 68 0 R] /Parent 3 0 R 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 /FontDescriptor 19 0 R /Kids [99 0 R 100 0 R 101 0 R 102 0 R 103 0 R 104 0 R] /Length 1692 /BaseFont/HVCESD+CMBX12 /S /GoTo Read Online Complex Analysis << Step 3: Add C. Example: ∫3x 5, dx. 19 0 obj chapter 05: sequences and series of complex numbers endobj /Parent 9 0 R >> 8 0 obj endobj /Kids [35 0 R 36 0 R] /Count 4 >> endobj 57 series problems with answers. 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