THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
11. THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
Although we have already defined e z by a power series (8.1), it is worth while to write it in another form. By (8.2) we can write
e z =e x+iy =e x e iy =e x (cos y + i sin y).
This is more convenient to use than the infinite series if we want values of e z for given z. For example,
e 2−iπ =e 2 e −iπ =e 2 · (−1) = −e 2
from Figure 9.2. We have already seen that there is a close relationship [Euler’s formula (9.3)] between complex exponentials and trigonometric functions of real angles. It is useful to write this relation in another form. We write Euler’s formula (9.3) as it
68 Complex Numbers Chapter 2
is, and also write it with θ replaced by −θ. Remember that cos(−θ) = cos θ and sin(−θ) = − sin θ. Then we have
e iθ = cos θ + i sin θ,
e −iθ = cos θ − i sin θ. These two equations can be solved for sin θ and cos θ. We get (Problem 2)
These formulas are useful in evaluating integrals since products of exponentials are easier to integrate than products of sines and cosines. (See Problems 11 to 16, and Chapter 7, Section 5.)
So far we have discussed only trigonometric functions of real angles. We could define sin z and cos z for complex z by their power series as we did for e z . We could then compare these series with the series for e iz and derive Euler’s formula and (11.3) with θ replaced by z. However, it is simpler to use the complex equations corresponding to (11.3) as our definitions for sin z and cos z. We define
The rest of the trigonometric functions of z are defined in the usual way in terms of these; for example, tan z = sin z/ cos z.
e i·i +e −i·i
e −1 +e
Example 1. cos i =
= 1.543 · · ·. (We will see in Section 15 that
this expression is called the hyperbolic cosine of 1.)
Example 2.
e i(π/2+i ln 2) −e −i(π/2+i ln 2)
e iπ/2 e − ln 2
−e
−iπ/2 e ln 2
by (8.2).
2i
From Figures 5.2 and 9.3, e iπ/2 = i, and e −iπ/2 = −i. By the definition of ln x [or see equations (13.1) and (13.2)], e ln 2 = 2, so e − ln 2 = 1/e ln 2 = 1/2. Then
+ i ln 2 = (i)(1/2) − (−i)(2) = .
sin
2 2i
Section 12 The Exponential and Trigonometric Functions
Notice from both these examples that sines and cosines of complex numbers may
be greater than 1. As we shall see (Section 15), although | sin x| ≤ 1 and | cos x| ≤ 1 for real x, when z is a complex number, sin z and cos z can have any value we like. Using the definitions (11.4) of sin z and cos z, you can show that the familiar trigonometric identities and calculus formulas hold when θ is replaced by z.
Example 3. Prove that sin 2 z + cos 2 z = 1.
e 2iz −2+e −2iz
e 2iz +2+e −2iz
cos z=
2 2 sin 2 z + cos 2 z= + = 1.
Example 4. Using the definitions (11.4), verify that (d/dz) sin z = cos z.
PROBLEMS, SECTION 11
1. Define sin z and cos z by their power series. Write the power series for e iz . By comparing these series obtain the definition (11.4) of sin z and cos z.
2. Solve the equations e iθ = cos θ + i sin θ, e −iθ = cos θ − i sin θ, for cos θ and sin θ and so obtain equations (11.3).
Find each of the following in rectangular form x + iy and check your results by computer. Remember to save time by doing as much as you can in your head.
5. e (iπ/4)+(ln 2)/2 6. cos(i ln 5)
3. e −(iπ/4)+ln 3
4. e 3 ln 2−iπ
8. cos(π − 2i ln 3) 9. sin(π − i ln 3)
7. tan(i ln 2)
10. sin(i ln i)
In the following integrals express the sines and cosines in exponential form and then integrate to show that:
Evaluate Re (a+ib)x dx and take real and imaginary parts to show that: Z
ax
17. e ax cos bx dx = e (a cos bx + b sin bx)
a 2 +b 2
18. e ax sin bx dx = e ax (a sin bx − b cos bx)
a 2 +b 2
70 Complex Numbers Chapter 2
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
Show more