THE COMPLEX PLANE
3. THE COMPLEX PLANE
In analytic geometry we plot the point (5, 3) as shown in Figure 3.1. As we have seen, the symbol (5, 3) could also mean the complex number 5 + 3i. The point (5, 3) may then be labeled either (5, 3) or 5 + 3i. Similarly, any complex number x + iy (x and y real) can be represented by a point (x, y) in the (x, y) plane. Also any point (x, y) in the (x, y) plane can be labeled x + iy as well as (x, y). When the (x, y)
48 Complex Numbers Chapter 2
plane is used in this way to plot complex numbers, it is called the complex plane. It is also sometimes called an Argand diagram. The x axis is called the real axis, and the y axis is called the imaginary axis (note, however, that you plot y and not iy).
Figure 3.1
When a complex number is written in the form x + iy, we say that it is in rectangular form because x and y are the rectangular coordinates of the point representing the number in the complex plane. In analytic geometry, we can locate
a point by giving its polar coordinates (r, θ) instead of its rectangular coordinates (x, y). There is a corresponding way to write any complex number. In Figure 3.2,
x = r cos θ, (3.1) y = r sin θ.
Then we have x + iy = r cos θ + ir sin θ
(3.2) Figure 3.2 = r (cos θ + i sin θ).
This last expression is called the polar form of the complex number. As we shall see (Sections 9 to 16), the expression (cos θ + i sin θ) can be written as e iθ , so a convenient way to write the polar form of a complex number is
x + iy = r(cos θ + i sin θ) = re iθ .
The polar form re iθ of a complex number is often simpler to use than the rectangu- lar form.
Example. In Figure 3.3 the point A could be √ √ labeled as (1,
3) or as 1 + i
3. Simi-
larly, using polar coordinates, the point
A could be labeled with its (r, θ) values as (2, π/3). Notice that r is always taken positive. Using (3.3) we have
1+i 3=2 cos + i sin
= 2e iπ/3 .
This gives two more ways to label point
A in Figure 3.3. Figure 3.3
Section 4 Terminology and Notation
Radians and Degrees In Figure 3.3, the angle π/3 is in radians. Ever since you studied calculus, you have been expected to measure angles in radians and not degrees. Do you know why? You have learned that (d/dx) sin x = cos x. This formula is not correct—unless x is in radians. (Look up the derivation in your calculus book!) Many of the formulas you now know and use are correct only if you use radian measure; consequently that is what you are usually advised to do. However, it is sometimes convenient to do computations with complex numbers using degrees, so it is important to know when you can and when you cannot use degrees. You can use degrees to measure an angle and to add and subtract angles as long as the final step is to find the sine, cosine, or tangent of the resulting angle (with your calculator in degree mode). For example, in Figure 3.3, we can, if we like, say that θ = 60 ◦ instead of θ = π/3. If we want to find sin(π/3−π/4) = sin(π/12) =
0.2588 (calculator in radian mode), we can instead find sin(60 ◦ − 45 ) = sin 15 = 0.2588 (calculator in degree mode). Note carefully that an angle is in radians unless ◦
the degree symbol is used; for example, in sin 2, the 2 is 2 radians or about 115 .
In formulas, however, use radians. For example, in using infinite series, we say that sin θ ∼ = θ for very small θ. Try this on your calculator; you will find that it is true in radian mode but not in degree mode. As another example, consider
0 dx/(1 + x 2 ) = arc tan 1 = π/4 = 0.785. Here arc tan 1 is not an angle; it is the numerical value of the integral, so the answer 45 (obtained from a calculator in
degree mode) is wrong! Do not use degree mode in reading an arc tan (or arc sin or arc cos) unless you are finding an angle [for example, in Figure 3.2, θ = arc tan(y/x), √ ◦
and in Figure 3.3, θ = arc tan
3 = π/3 or 60 ].
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
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