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Introduction Polar form Exponential form Applications Euler’s formula Test POLAR AND EXPONENTIAL FORMS ALGEBRA 7 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Polar and exponential forms 1 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Objectives This presentation will cover the following: • Argand diagrams and the complex plane • Modulus and argument • Conversion between Cartesian, polar and exponential form • Logarithm of a complex number This will prepare us for later topics such as De Moivre’s theorem. Polar and exponential forms 2 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Argand diagram We’ve previously seen that complex numbers of the form x + yi are said to be in Cartesian form because they can be represented as points (x, y) on a graph called an Argand diagram. yi 4 z3 z1 = 4 + 3i z2 z2 = 4i z1 3 z3 = −2 + 3i 2 z4 = −4 1 z5 = 3 − 2i z4 x −4 −3 −2 −1 −1 1 2 3 4 z5 −2 This means we can examine complex numbers using the rules of geometry, trigonometry and algebra. Polar and exponential forms 3 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Polar form of a complex number Argand diagrams show us how to express complex numbers in polar form. yi A complex number in polar form is defined by its modulus (length) and argument (angle). z = x + yi The modulus is the distance r of the point from the origin. r The argument is an angle θ θ x measured anticlockwise from the positive In polar form x-axis.the complex number z = x + yi becomes y x z = r cos θ + (r sin θ )i The modulus r and argument θ is given by y p r = x2 + y 2 and θ = tan−1 x and θ must be in the correct quadrant. Polar and exponential forms 4 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Cartesian to polar form Express z = −4 + i in polar form. Give the argument, in radians to 3 S.F. In this case we have |z| = p 17 . Since z is in the second quadrant then arg z = π + tan−1 (− 41 ) = 2.897 (radians). Therefore z= p 17(cos 2.897 + i sin 2.897) A quick way of representing the polar form is to use the r cis θ or r∠θ notations. p p For example z = 17 cis 2.897 or z = 17∠2.897. Polar and exponential forms 5 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Polar to cartesian form Express z = 10∠ 7π 6 in cartesian form. In this case we have |z| = 10 and θ = 7π 6 . Therefore 7π z = 10(cos 7π 6 + i sin 6 ) p = 10(− 23 + i(− 21 )) p ∴ z = −5 3 − 5i Polar and exponential forms 6 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Exponential form Consider the Maclaurin series for the exponential function: ex = 1 + x + x2 x3 x4 x5 + + + ... 2! 3! 4! 5! Substitute x = iθ to get a series for eiθ eiθ = 1 + iθ + (iθ )2 (iθ )3 (iθ )4 (iθ )5 + + + ... 2! 3! 4! 5! Expanding the brackets and recalling that i2 = −1, i3 = −i, etc: eiθ Polar and exponential forms i2 θ 2 i3 θ 3 i4 θ 4 i5 θ 5 + + + ... 2! 3! 4! 5! θ 2 iθ 3 θ 4 iθ 5 − + + ... = 1 + iθ − 2! 3! 4! 5! = 1 + iθ + 7 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Grouping the real and imaginary terms we get: θ2 θ4 θ3 θ5 iθ e = 1− + −... +i θ − + ... 2! 4! 3! 5! The terms within the first set of brackets are the terms for the infinite series for cos θ . The terms within the second set of brackets are the terms of the infinite series for the function sin θ . Therefore: eiθ = cos θ + i sin θ This relationship is known as Euler’s Identity. If we multiply both sides of this by the modulus r we obtain a slightly more general form: reiθ = r(cos θ + i sin θ ) Polar and exponential forms 8 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Cartesian to polar and exponential form p Express z = 3 − i in polar and exponential form. The modulus is |z| = qp ( 3)2 + (−1)2 = 2 The argument is in the fourth quadrant so that −1 arg z = tan( p ) + 2π = 3 11π 6 The polar form of the number is 11π z = 2(cos 11π 6 + i sin 6 ) and the exponential form is z = 2ei Polar and exponential forms 11π 6 9 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Powers Powers of complex numbers Changing a complex number to exponential or polar form makes it easy to evaluate powers of that number. Evaluating a power Evaluate (1 − i)4 . We can write 1 − i in exponential form as p 7π 1 − i = 2e 4 i Raising both sides to the power of 4: (1 − i)4 = p 7π ( 2e 4 i )4 = 4e7πi Changing back to cartesian form (via polar form): (1 − i)4 4 ∴ (1 − i) = 4(cos 7π + i sin 7π) = 4(−1) = −4 For higher powers (or nth roots) it may be necessary to use De Moivre’s theorem. We’ll study this method at a later time. Polar and exponential forms 10 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Logarithms Logarithms of complex numbers The function ln x for x ∈ R is defined on the domain x > 0, but we can extend the domain to x ∈ C. Complex logarithm p Evaluate ln(1 + 3i). p Express 1 + 3i in exponential form. Here we have r = 2 and θ = 13 π. Therefore π p ln(1 + 3) = ln 2ei 3 Use log rules to simplify: π = ln 2 + ln ei 3 p ln(1 + 3) = ln 2 + i π3 Polar and exponential forms 11 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Logarithms Multivalued functions In the previous example the angle calculated was the principal value: π 3. We could also have chosen θ = 31 π + 2π = 37 π to represent the same logarithm value. In general, the complex logarithm can take the values specified by p n∈Z ln(x + yi) = ln x2 + y 2 + arg(x + yi) + 2nπ, Although the exponential (and polar) forms differ by multiples of 2π — the Cartesian forms are all identical. The complex logarithm is sometimes called “a multivalued function”. As we saw in Semester 1, functions are supposed to have one input and one output so this name is a misnomer! We will return to this concept with De Moivre’s theorem in the final part of the course. Other examples of multivalued functions are square-roots, in which every real number is associated with two roots. Polar and exponential forms 12 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Finally...a beautiful equation! Given the relationship reiθ = r(cos θ + i sin θ ) If we put r = 1 and θ = π into we obtain (1)eiπ = eiπ (1)(cos π + i sin π) ⇒ eiπ = −1 Rearranging we can make this equation eiπ + 1 = 0 This is called Euler’s Formula and it is often cited as being one of the most beautiful discoveries in all of mathematics. It relates, in a single equation, five of the most important numbers in mathematics. Polar and exponential forms 13 / 14 Adrian Jannetta Introduction Polar form Exponential form Applications Euler’s formula Test Test yourself... Given the complex numbers z1 = 5i z2 = −5 − 12i z3 = 10ei 3π 4 1 Find |z1 | and arg z1 2 Express z2 in polar and exponential form. 3 Express z3 in Cartesian form. 4 Calculate z2 z3 in polar form 5 Calculate ln z3 6 Calculate ln(−2) Answers: 1 |z1 | = 5 and arg z1 = π 2 2 z2 = 13(cos 4.318 + i sin 4.318) and z = 13e4.318i p p 3 z3 = −5 2 + 5 2i Polar and exponential forms 4 z2 z3 = (13)(10)ei(4.318+ 3π 4 ) = 130e6.674i 5 ln z3 = ln 10 + 34 πi 6 −2 = 2eiπ . Therefore ln(−2) = ln 2 + πi 14 / 14 Adrian Jannetta