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CALCULUS 3Chapter 1COMPLEX NUMBERSAND FUNCTIONSAssoc. Prof. N. Dinh, Assoc. Prof. N. N. HaiDEPARTMENT OF MATHEMATICSINTERNATIONAL UNIVERSITY, VNU-HCMJuly 11, 2018Assoc. Prof. N. Dinh, Assoc. Prof. N. N. HaiCALCULUS 3Chapter 1COMPLEX NUMBERSAND FUNCTIONS
ReferencesMain textbook:G. James,Advanced Modern EngineeringMathematics, 4thed., Prentice Hall, 2011.Other textbooks:G. James,Modern Engineering Mathematics, 3rded., Prentice Hall, 2004.E. Kreyszig,Advanced Engineering Mathematics,10thed., John Wiley & Sons, 2011.Assoc. Prof. N. Dinh, Assoc. Prof. N. N. HaiCALCULUS 3Chapter 1COMPLEX NUMBERSAND FUNCTIONS
Chapter 1COMPLEX NUMBERS ANDCOMPLEX FUNCTIONS
Assoc. Prof. N. Dinh, Assoc. Prof. N. N. HaiCALCULUS 3Chapter 1COMPLEX NUMBERSAND FUNCTIONS
1COMPLEX NUMBERS1.1DEFINITION OF COMPLEX NUMBERSThe symboljthat has the propertyj2=-1iscalledthe imaginary unit. We could also calljthesquare root of-1,j=-1. Of coursejisNOTareal number.1.1DEFINITION OF COMPLEX NUMBERSDefinition 1.1Acomplex numberis an expression of the forma+jbora+bjwhereaandbarereal numbers, andjis theimaginary unit.Assoc. Prof. N. Dinh, Assoc. Prof. N. N. HaiCALCULUS 3Chapter 1COMPLEX NUMBERSAND FUNCTIONS
1COMPLEX NUMBERS1.1DEFINITION OF COMPLEX NUMBERSThereal partof the complex numberz=a+bjisthe real numberaand denoted Re(z). We call thereal numberbthe imaginary partofzand denoteit Im(z). So a complex number is the sum of twoterms:a complex number= Real part+j(Imaginary part)Assoc. Prof. N. Dinh, Assoc. Prof. N. N. HaiCALCULUS 3Chapter 1COMPLEX NUMBERSAND FUNCTIONS
1COMPLEX NUMBERS1.1DEFINITION OF COMPLEX NUMBERSExample 1.1.Re(2 +3j) = 2Im(2 +3j) =3Re(-4j) = Re(0 + (-4)j)= 0Im(-4j) =-4Ifa= 0, the complex numberz=bjis said to bepurely imaginary, and ifb= 0, the complexnumberz=ais said to bepurely real.Assoc. Prof. N. Dinh, Assoc. Prof. N. N. HaiCALCULUS 3Chapter 1COMPLEX NUMBERSAND FUNCTIONS
1COMPLEX NUMBERS1.2ARITHMETICAL OPERATIONSComplex numbers have many applications inengineering. To use them, we must know how tocarry out the usual arithmetical operations.EqualityDefinition 1.2Two complex numbers areequalif their real partsare equal and their imaginary parts are equal.a+bj=c+dj⇐⇒a=candb=dThus when two complex numbers are equal wecan equate their respective real and imaginary parts.Assoc. Prof. N. Dinh, Assoc. Prof. N. N. HaiCALCULUS 3Chapter 1COMPLEX NUMBERSAND FUNCTIONS
1COMPLEX NUMBERS1.2ARITHMETICAL OPERATIONSAddition and SubtractionDefinition 1.3Ifz1=a+bjandz2=c+dj, then we definez1+z2= (a+c) + (b+d)jz1-z2= (a-c) + (b-d)jThat is, the sum and difference of two complexnumbers are defined byadding or subtracting their

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