/Type /Pages /Name/F1 /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 /FirstChar 33 << 10 questions on geometric series, sequences, and l'Hôpital's rule with answers. >> >> /Type /Pages /Kids [123 0 R 124 0 R 125 0 R 126 0 R 127 0 R 128 0 R] /Kids [129 0 R 130 0 R 131 0 R 132 0 R 133 0 R 134 0 R] /Kids [117 0 R 118 0 R 119 0 R 120 0 R 121 0 R 122 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. 17 0 obj >> 323.4 354.2 600.2 323.4 938.5 631 569.4 631 600.2 446.4 452.6 446.4 631 600.2 815.5 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 0 0 0 0 0 0 0 693.8 954.4 868.9 /LastChar 196 31 0 obj 877 0 0 815.5 677.6 646.8 646.8 970.2 970.2 323.4 354.2 569.4 569.4 569.4 569.4 569.4 /Parent 8 0 R 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 /Name/F2 endobj endobj 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 /Type /Pages endobj 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 /PTEX.Fullbanner (This is pdfTeX, Version 3.14159265-2.6-1.40.16 \(TeX Live 2015\) kpathsea version 6.2.1) << All you need to know are the rules that apply and how different functions integrate. endobj << Integration reverse of differentiation questions and worked. << 161/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus /Type/Encoding << << /Type /Pages /Kids [135 0 R 136 0 R 137 0 R 138 0 R 139 0 R] . /Type /Pages questions about Taylor series with answers. << 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 I have done my best to ensure that the solutions are clear and correct, and that the level of rigor is at least as high as that /Keywords () /Count 36 harmonic functions provided by the real and imaginary parts of the complex function are indeed solutions to the two-dimensional Laplace equation. 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 /Count 6 If values of three variables are known, then the others can be calculated using the equations. 22 0 obj /FontDescriptor 26 0 R /Parent 9 0 R 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 /D (chapter*.2) We need some more (easy!) 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 << 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 The various types of functions you will most commonly see are mono… 4 0 obj >> Fall 02-03 midterm with answers. endobj >> /Name/F4 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 26 0 obj >> endobj >> /Title (Foreword) 24 0 obj 7.1 Contour Integration: The complex integration along the scro curve used in evaluating the de nite integral is called contour integration. 14 0 obj /Count 3 For instance, complex functions are necessarily analytic, /Count 6 endobj /Count 6 Spring 03 midterm with answers. /Prev 34 0 R /Parent 3 0 R 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 >> /Parent 8 0 R /Names 4 0 R 13 0 obj COMPLEX INTEGRATION Example: Consider the diﬀerential form zm dz for integer m 6= 1. /Parent 8 0 R endobj It also connects widely with other branches of mathematics. >> 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 /Parent 2 0 R /BaseFont/QXVOCG+CMR7 /Parent 14 0 R 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 594.7 542 557.1 557.3 668.8 404.2 472.7 607.3 361.3 1013.7 706.2 563.9 588.9 523.6 >> /Type /Pages endobj /Type /Pages /Count 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 706.4 938.5 877 781.8 754 843.3 815.5 877 815.5 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 /Count 6 35 0 obj stream /Last 147 0 R 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] It is exact, since zm dz = 1 m+1 dzm+1. << Today we'll learn more about complex integration, we'll look at some examples, and we'll learn some first facts. 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 6 Integration: to solve complex environmental problems unintended negative consequences, or create new environmental or socio-economic problems12. >> /OpenAction 5 0 R /LastChar 196 << 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 endobj For example, establishing monoculture plantations to sequester carbon could diminish biological diversity and downstream water availability, and affect diets and nutrition13. /Count 6 /FontDescriptor 9 0 R << /LastChar 196 33 0 obj 24 0 obj /Count 6 323.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 323.4 323.4 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] /ModDate (D:20161215200015+10'00') /D (Item.259) << endobj /F 2 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. chapter 01: complex numbers, introductory remarks. 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 28 0 obj /Kids [93 0 R 94 0 R 95 0 R 96 0 R 97 0 R 98 0 R] 21 0 obj >> << endobj INTEGRATION PRACTICE QUESTIONS WITH SOLUTIONS. %���� /Next 11 0 R How to derive the rule for Integration by Parts from the Product Rule for differentiation, What is the formula for Integration by Parts, Integration by Parts Examples, Examples and step by step Solutions, How to use the LIATE mnemonic for choosing u and dv in integration by parts /FontDescriptor 23 0 R >> << endobj /Kids [20 0 R 21 0 R 22 0 R 23 0 R 24 0 R 25 0 R] Complex Integration ( Part 2 ) Explanation & Examples - When the contour is a straight line or a parabola Thank you guys for watching. /Kids [51 0 R 52 0 R 53 0 R 54 0 R 55 0 R 56 0 R] >> /Kids [63 0 R 64 0 R 65 0 R 66 0 R 67 0 R 68 0 R] 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 /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/exclam/quotedblright/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/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/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/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/endash/emdash/hungarumlaut/tilde/dieresis/suppress Numbers, Functions, Complex Integrals and Series. /BaseFont/VYRNZU+CMMI7 27 0 obj endobj >> 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 Complex Integration 6.1 Complex Integrals In Chapter 3 we saw how the derivative of a complex function is defined. << /Kids [35 0 R 36 0 R] << endobj << Solution The path of integration has length L = 4π. /Type/Font /Kids [148 0 R 149 0 R 150 0 R 151 0 R 152 0 R 153 0 R] << Quadratic Equations with Complex Solutions. /Pages 2 0 R /Kids [99 0 R 100 0 R 101 0 R 102 0 R 103 0 R 104 0 R] 6 0 obj 16 0 obj << 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 /Last 143 0 R (pdf) complex analysis: problems with solutions. endobj /Filter[/FlateDecode] Here we are going to see under three types. /Count 4 endobj >> /Type /Pages /Type /Page Writing z = x + iy, we have |ez| = |ex+iy| = ex ≤ e2, for … /Kids [26 0 R 27 0 R 28 0 R 29 0 R 30 0 R] 10 0 obj /Limits [(Doc-Start) (Item.56)] << /Encoding 21 0 R /Count 5 /Type /Pages /S /GoTo I will be grateful to everyone who points out any typos, incorrect solutions, or sends any other /Type /Pages When m ≥ 0 this is deﬁned in the entire complex plane; when m < 0 it is deﬁned in the punctured plane (the plane with 0 removed). endobj /CreationDate (D:20161215200015+10'00') /Type/Encoding endobj /Count 6 11 0 obj Often solutions to quadratic equations are not real. /Type/Font /Encoding 7 0 R /Parent 9 0 R /BaseFont/HVCESD+CMBX12 We will then discuss complex integration, culminating with the /FontDescriptor 12 0 R 12 0 obj >> >> 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 >> Branch Cut Integration Complex Integration Contour Integrals Examples and Solutions in Complex Integration Hypergeometric Function Undergraduate Course on Complex Integration Wiener-Hopf Equation . 2 0 obj 49 integration problems with answers. /Subtype/Type1 truth! >> /Parent 8 0 R 15 0 obj Write x+ i x− i = x+i x−i × x+i x+i = x2 +2ix− 1 x2 +1 = (x2 +1)+2ix−2 x2 +1 =1− 2 x2 +1 + 2ix x2 +1. << /Parent 8 0 R endobj We'll start by introducing the complex plane along with the algebra and geometry of complex numbers and make our way via differentiation, integration, complex dynamics and power series representation into territories at the edge of what's known today. /FontDescriptor 19 0 R /Kids [14 0 R 15 0 R 16 0 R 17 0 R 18 0 R 19 0 R] 874 706.4 1027.8 843.3 877 767.9 877 829.4 631 815.5 843.3 843.3 1150.8 843.3 843.3 /Count 6 756 339.3] /Name/F6 endobj 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 32 0 obj Enterprise integration patterns solving integration problems using. endobj /Count 6 << /Encoding 17 0 R /Parent 3 0 R 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 /Outlines 3 0 R 18 0 obj 692.5 323.4 569.4 323.4 569.4 323.4 323.4 569.4 631 507.9 631 507.9 354.2 569.4 631 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 Kinematic equations relate the variables of motion to one another. /Parent 9 0 R /D [13 0 R /Fit] 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. << 500 500 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 0 0 0 0 0 0 0 625 833.3 7 Evaluation of real de nite Integrals as contour integrals. In fact, to a large extent complex analysis is the study of analytic functions. x��YKs�6��W�HM"�x3�x�M�Lgz�gr�{`dڢ+��Dŉ}w>@Td'mO�`��[email protected]�,�M�����W4aQ*��I� F%K� �2�|�g��:�X�Œk���_����h��d))�ϭ�?n�/~n�]�,���]^�ն]I�]i �n%%t����P�L�������|�Ro�L?�G/�%�Xg;e��d ���)ɯ��e�4x�4'���w%h*o�z9. /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 /Next 32 0 R /Kids [105 0 R 106 0 R 107 0 R 108 0 R 109 0 R 110 0 R] Of course, no project such as this can be free from errors and incompleteness. /Producer (pdfTeX-1.40.16) contents: complex variables . 160/space/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 173/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis] 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). 57 series problems with answers. Furthermore, a substitution which at ﬁrst sight might seem sensible, can lead nowhere. /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 /BaseFont/GDTASL+CMR10 20 0 obj endobj (1.17) On the other hand, the diﬀerential form dz/z is closed but not exact in the punctured plane. Example 9: Solve using the quadratic formula: x 2 − 2 x + 5 = 0. Indefinite Integrals, Step By Step Examples. endobj /Kids [154 0 R 155 0 R 156 0 R 157 0 R 158 0 R 159 0 R] /Kids [81 0 R 82 0 R 83 0 R 84 0 R 85 0 R 86 0 R] This is for questions about integration methods that use results from complex analysis and their applications. /Type /Pages /Parent 2 0 R /Type /Catalog /Parent 7 0 R /Count 6 endobj << /LastChar 196 500 555.6 527.8 391.7 394.4 388.9 555.6 527.8 722.2 527.8 527.8 444.4 500 1000 500 course. /Subtype/Type1 /A 144 0 R 8 0 obj /Subject () 30 0 obj endobj >> /rgid (PB:280722238_AS:[email protected]) >> endobj /BaseFont/QCGQLN+CMMI10 Solutions to integration by parts. You know the problem is an integration problem when you see the following symbol: Remember, too, that your integration answer will always have a constant of integration, which means that you are going to add '+ C' for all your answers.

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