/FirstChar 33 594.7 542 557.1 557.3 668.8 404.2 472.7 607.3 361.3 1013.7 706.2 563.9 588.9 523.6 The complex exponential The exponential function is a basic building block for solutions of ODEs. xref In order to compute E1(z) olltsid e this range, (or within this 21 0 obj 0000007401 00000 n 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 >> complex exponential. 0000006765 00000 n Integrals Producing Logarithmic Functions. In this view, the x axis is the real part 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] 485 0 obj <> endobj 0000064868 00000 n Tables of the Exponential Integral Ei(x) In some molecular structure calculations it is desirable to have values of the integral Ei(s) to higher accuracy than is provided by the standard tables [1} /LastChar 196 MAGIC WITH COMPLEX EXPONENTIALS 99 It is useful to think about a complex number as being a vector in a two dimensional space, as in Fig. This table covers the range Ixl ~ 20, Iyl ~ 20, with argumcnts variously spaced. 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] Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. 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 0 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 4. Applications of the Complex Exponential Integral By Murlan S. Corrington 1. 2. /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 endobj 0000047190 00000 n 0000057058 00000 n The exponential integrals,,,,,, and are defined for all complex values of the parameter and the variable. integrals, which can be used to obtain integrals not presented in this book. 10 0 obj 0000047504 00000 n 0000048332 00000 n Fifth edition, 1994 Table of contents ... 6.2-6.3 The Exponential-Integral Function and Functions Generated by it ... 11.31 Inequalities for sets of complex numbers 12 Integral Inequalities 12.1-12.5 Properties of Integrals and 0000064457 00000 n 0000016203 00000 n 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 0000002240 00000 n The following problems involve the integration of exponential functions. This section is the table of Laplace Transforms that we’ll be using in the material. The real root of the exponential integral occurs at 0.37250741078... (OEIS A091723 ), which is , where is Soldner's constant (Finch 2003). >> (Challenging) Factoring z2 + 1 = (z + i)(z ¡ i) and using partial fractions, integrate (formally) Z 1 z2 +1 dz and try to get back to the arctan you know and love by using the complex … << last integral. << 0000032031 00000 n 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 0000006158 00000 n 2.3. /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 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 29 0 obj 24 0 obj 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 << >> 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 0 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 tulane. H�\��n�@E���Y���6�{�H� �*T�D���;�A��m���w� �M���}�����mی&�zF����]�:�C86mV.�o���nZ�S�gy:y�z�i��άVY��6���j��7���7������!f����дGs����� The exponential integral EnHzL is connected with the inverse of the regularized incomplete gamma function Q-1Ha,zL by the following formula: EnIQ-1H1-n,zLM−Q-1H1-n,zL n-1 GH1-nLz. 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 756 339.3] /FontDescriptor 15 0 R /LastChar 196 /LastChar 196 To improve this 'Exponential integral Ei(x) Calculator', please fill in questionnaire. /LastChar 196 0000007444 00000 n 0000001444 00000 n ����N�M1��z����gu >> The product of two integrals can be expressed as a double integral: I2 = Z ∞ −∞ Z ∞ −∞ e−(x2+y2) dxdy The differential dxdy represents an elementof area in cartesian coordinates, with the domain of integration extending over the entire xy-plane. 0000019067 00000 n 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 Male or Female ? /FirstChar 33 /FirstChar 33 0000007499 00000 n 0000002052 00000 n /BaseFont/QXVOCG+CMR7 /Name/F3 endobj 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 Signals & Systems - Reference Tables 4 Some Useful Mathematical Relationships 2 cos( ) ejx e jx x j e e x jx jx 2 sin( ) cos(x y) cos(x)cos(y) sin(x)sin(y) ... Reference Tables 5 Useful Integrals cos(x)dx sin(x) sin(x)dx cos(x) 0000061895 00000 n 0000062528 00000 n For fixed, the exponential integral is an entire function of. /BaseFont/HVCESD+CMBX12 The recent publication of an extensive table of the exponential integral for complex arguments makes it possible to evaluate a large number of indefinite integrals not in existing tables, and to obtain values for the sine and cosine integrals for complex arguments. /Name/F6 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 /Subtype/Type1 Table of Integrals, Series, and Products. 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 0000067844 00000 n /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/alpha/beta/gamma/delta/epsilon1/zeta/eta/theta/iota/kappa/lambda/mu/nu/xi/pi/rho/sigma/tau/upsilon/phi/chi/psi/omega/epsilon/theta1/pi1/rho1/sigma1/phi1/arrowlefttophalf/arrowleftbothalf/arrowrighttophalf/arrowrightbothalf/arrowhookleft/arrowhookright/triangleright/triangleleft/zerooldstyle/oneoldstyle/twooldstyle/threeoldstyle/fouroldstyle/fiveoldstyle/sixoldstyle/sevenoldstyle/eightoldstyle/nineoldstyle/period/comma/less/slash/greater/star/partialdiff/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/flat/natural/sharp/slurbelow/slurabove/lscript/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/dotlessi/dotlessj/weierstrass/vector/tie/psi Integrals of Exponential Functions; Integrals Involving Logarithmic Functions; Key Concepts. 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 17 0 obj %%EOF 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 /Type/Font Complex numbers expand the scope of the exponential function, and bring trigonometric functions under its sway. 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 Euler’s formula defines the exponential to a pure imaginary power. 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 /Subtype/Type1 /Length 1692 161/minus/periodcentered/multiply/asteriskmath/divide/diamondmath/plusminus/minusplus/circleplus/circleminus %PDF-1.4 %���� 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 (1.1) It is said to be exact in … Published 1940 /Subtype/Type1 �ʌ�22�|� �����s[4�غ8��'�6��¤&I�����O\�� /BaseFont/VYRNZU+CMMI7 William Vernon Lovitt, Linear Integral Equations, McGraw-Hill Book Co., Inc., New York, 1924. National Bureau of Standards. 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 /Subtype/Type1 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 Contributors; Exponential and logarithmic functions are used to model population growth, cell growth, and financial growth, as well as depreciation, radioactive decay, and resource consumption, to name only a few applications. 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 27 0 obj π: the ratio of the circumference of a circle to its diameter, ∈: element of, e: base of natural logarithm, E 1 ⁡ (z): exponential integral, i: imaginary unit, ℤ: set of all integers and z: complex variable 0000058344 00000 n 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 0 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] jsp) • V. H. Moll, The Integrals in Gradshteyn and Ryzhik (http:/ / www. 0000059052 00000 n /FontDescriptor 26 0 R A simple table of derivatives and integrals from the Gottfried Leibniz archive. The definition of an exponential to an arbitrary complex power is: ea+ib= eaeib= ea(cos(b)+ i sin(b)). The recent publication of an extensive table of the exponential integral for complex arguments [1] makes it possible to evaluate a large number of indefinite integrals not in existing tables… math. x��YKs�6��W�HM"�x3�x�M�Lgz�gr�{`dڢ+��Dʼn}w>@Td'mO�`��~@IF�,�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. <<2BFCD845482BB74EAEF6E5938D54D746>]>> /Encoding 17 0 R 173/circlemultiply/circledivide/circledot/circlecopyrt/openbullet/bullet/equivasymptotic/equivalence/reflexsubset/reflexsuperset/lessequal/greaterequal/precedesequal/followsequal/similar/approxequal/propersubset/propersuperset/lessmuch/greatermuch/precedes/follows/arrowleft/spade] 540 0 obj<>stream 0000003299 00000 n Leibniz developed integral calculus at around the same time as Isaac Newton. /BaseFont/GDTASL+CMR10 /LastChar 196 /Type/Font 0000007527 00000 n Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has Author United States. 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 << 0000031706 00000 n /Name/F2 π: the ratio of the circumference of a circle to its diameter, ∈: element of, e: base of natural logarithm, E 1 ⁡ (z): exponential integral, i: imaginary unit, ℤ: set of all integers and z: complex variable /Type/Font /Name/F1 endobj %PDF-1.2 /BaseFont/QCGQLN+CMMI10 The integral table in the frame above was produced TeX4ht for MathJax using the command sh ./makejax.sh integral-table the configuration file here, and the shell scripts ht5mjlatex and makejax.sh 0000016799 00000 n COMPLEX INTEGRATION 1.2 Complex functions 1.2.1 Closed and exact forms In the following a region will refer to an open subset of the plane. wolfram. /Subtype/Type1 x�b```b``{������� �� @1v�Ǿ�r�1k3�ղ-cS``X�kѼ�Ā����{x8�5��� pV�aQ�ɔ;\ߡU���]N�O��(xHvg�P��vFƪR��+xC��궷Ѣ:�J,�� >> & >` �{�� /Widths[323.4 569.4 938.5 569.4 938.5 877 323.4 446.4 446.4 569.4 877 323.4 384.9 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 0000005121 00000 n 0000007611 00000 n << Comments. 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 endstream endobj 539 0 obj<>/Size 485/Type/XRef>>stream 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 endobj << Integrals of exponential functions. Evaluation of the exponential integral for large complex arguments @article{Todd1954EvaluationOT, title={Evaluation of the exponential integral for large complex arguments}, author={John Todd}, journal={Journal of research of the National Bureau of Standards}, year={1954}, volume={52}, pages={313} } This page lists some of the most common antiderivatives 0000002501 00000 n 339.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 585.3 339.3 The function is an analytical functions of and over the whole complex ‐ and ‐planes excluding the branch cut on the ‐plane. Since the derivative of ex is e x;e is an antiderivative of ex:Thus Z exdx= ex+ c Recall that the exponential function with base ax can be represented with the base eas elnax = com/ index. /Encoding 7 0 R A differential form pdx+qdy is said to be closed in a region R if throughout the region ∂q ∂x = ∂p ∂y. Integrals of Exponential and Trigonometric Functions. /Encoding 21 0 R /Type/Encoding x�bb�g`b``Ń3� ���ţ�1�1@� �� << A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity. Integrals with Trigonometric Functions Z sinaxdx= 1 a cosax (63) Z sin2 axdx= x 2 sin2ax 4a (64) Z sinn axdx= 1 a cosax 2F 1 1 2; 1 n 2; 3 2;cos2 ax (65) Z sin3 axdx= 3cosax 4a + cos3ax 12a (66) Z cosaxdx= endobj /Subtype/Type1 Do it also for ¡i and check that p ¡i = p ¡1 p i: 3. The copyright holder makes no representation about the accuracy, correctness, or 0 485 56 323.4 877 538.7 538.7 877 843.3 798.6 815.5 860.1 767.9 737.1 883.9 843.3 412.7 583.3 The quantity (OEIS A073003 ) is known as the Gompertz constant . A crazy notion: find ii by writing i as a complex exponential. stream /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 The first variable given corresponds to the outermost integral and is done last. 0000056468 00000 n 692.5 323.4 569.4 323.4 569.4 323.4 323.4 569.4 631 507.9 631 507.9 354.2 569.4 631 588.6 544.1 422.8 668.8 677.6 694.6 572.8 519.8 668 592.7 662 526.8 632.9 686.9 713.8 Definite integrals with finite limits are presented in the Part 2 only in the case when there are no corresponding indefinite integrals. /FontDescriptor 12 0 R 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 160/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] >> 0000002376 00000 n 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] 20 0 obj [Image source] Complex Numbers and the Complex Exponential 1. /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 0000063607 00000 n 0000055384 00000 n Key Equations. Improper integrals are presented independently of whether the corresponding indefinite integrals are presented or not. 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 843.3 507.9 569.4 815.5 877 569.4 1013.9 1136.9 877 323.4 569.4] Leibniz's table of derivatives and integrals. 0000059435 00000 n 0000042284 00000 n 0000025351 00000 n List of integrals of exponential functions 3 ( is the modified Bessel function of the first kind) References • Wolfram Mathematica Online Integrator (http:/ / integrals. 7 0 obj /FirstChar 33 /Encoding 7 0 R 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 /Name/F5 /Name/F4 >> ©2005 BE Shapiro Page 3 This document may not be reproduced, posted or published without permission. /Encoding 17 0 R In mathematics, the exponential integral Ei is a special function on the complex plane.It is defined as one particular definite integral of the ratio between an exponential function and its argument. �d�$��r�Ln�R���M8�8M@ѥ���W���6]=}|Н!�t:�(�fG��ơ�(^fRec�#P�� DH��=Ęь%%���XZ��Gz� �,�@����"2|�-��]�9�HM�fr�l`��v��ᑸC��2�Kݸ��4x9��8��A���>�N0Y��,�k2�8��ac����L�\>b�6�+�P0�i�� �{�.,�G��4*5�2�0&*5 ;Y��q�=�w�>pQ}���@����@������PJ4c`|� 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 2.2. Multiple integrals use a variant of the standard iterator notation. Exponential solutions. endobj 16 0 obj Male Female Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation Elementary school/ Junior high-school student 0000055870 00000 n 0000025705 00000 n 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 >> 0000068469 00000 n 0000000016 00000 n 13 0 obj /LastChar 196 323.4 354.2 600.2 323.4 938.5 631 569.4 631 600.2 446.4 452.6 446.4 631 600.2 815.5 0000061060 00000 n >> 0000041148 00000 n html) These formulas lead immediately to the following indefinite integrals : 0000032739 00000 n 277.8 500] Indefinite integrals are antiderivative functions. 0000005574 00000 n DOI: 10.6028/JRES.052.045 Corpus ID: 6181894. 6. An extensive table of the exponential integral has been prepared by the National Bureau of Standards [1]; 1 the introduction to the table gives a precise definition of this function. << 0000063215 00000 n 0000057649 00000 n 0000004454 00000 n 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 Definition of Exponential Integral. 0000002874 00000 n 0000019545 00000 n 4. endobj (1) We stress that the equation (1) is a definition, not a self-evident truth, since up to now no meaning has been assigned to the left-hand side. Introduction. /FirstChar 33 /Type/Font The function et is defined to be the so­ lution of the initial value problem x˙ = x, x(0) = 1. /Type/Encoding 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 Complex Exponential Fourier Series T j nt n n j nt n f t e dt T f t F e F 0 0 1 ( ) , where . » Integrate can evaluate integrals of rational functions. The function $ \mathop{\rm Ei} $ is usually called the exponential integral. We give as wide a variety of Laplace transforms as possible including some that aren’t often given in tables of Laplace transforms. 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 500 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 /Widths[719.7 539.7 689.9 950 592.7 439.2 751.4 1138.9 1138.9 1138.9 1138.9 339.3 trailer startxref endstream endobj 486 0 obj<>/Metadata 53 0 R/AcroForm 487 0 R/Pages 52 0 R/StructTreeRoot 55 0 R/Type/Catalog/Lang(EN)>> endobj 487 0 obj<>/Encoding<>>>>> endobj 488 0 obj<>/ProcSet[/PDF/Text/ImageB]>>/Type/Page>> endobj 489 0 obj<> endobj 490 0 obj<> endobj 491 0 obj<> endobj 492 0 obj<> endobj 493 0 obj<>stream endobj Computation Laboratory. endobj 0000048928 00000 n /BaseFont/DIPVPJ+CMSY10 0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 /FontDescriptor 9 0 R %���� /Type/Font /FirstChar 33 /Filter[/FlateDecode] 0000061615 00000 n 0000026486 00000 n /Type/Font /Type/Encoding 1074.4 936.9 671.5 778.4 462.3 462.3 462.3 1138.9 1138.9 478.2 619.7 502.4 510.5 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 339.3 892.9 585.3 892.9 585.3 610.1 859.1 863.2 819.4 934.1 838.7 724.5 889.4 935.6 0000041543 00000 n /FontDescriptor 19 0 R 0000067178 00000 n /Encoding 7 0 R 0000033330 00000 n 0000018807 00000 n /FontDescriptor 23 0 R << 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 0 797.6 844.5 935.6 886.3 677.6 769.8 716.9 0 0 880 742.7 647.8 600.1 519.2 476.1 519.8 /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 6.1. >> We will assume knowledge of the following well-known differentiation formulas : , where , and , where a is any positive constant not equal to 1 and is the natural (base e) logarithm of a. << edu/ ~vhm/ Table. &��]Ӧ1�|;u�ù��0T�1d�e�6+��,��Ӟ�b>����ǴE:N��c� ���&�. 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