Index

Successive Approximations Example:
Real and Complex Solutions of w2Cos(w) + 3e-wSin(w) = w + 9

In this example, we'll find real and complex solutions of an equation containing trigonometric and exponential functions,

w2Cos(w) + 3e-wSin(w) = w + 9

First we'll subtract (w + 9) from both sides to get an equation equal to zero:

w2Cos(w) + 3e-w Sin(w) - w - 9 = 0

We'll compute real approximations of w with the function w ~ z - F(z)/S(z), where

z = a previous approximation of w, either guessed or computed
F(z) = an unrestricted function based on the equation = z2Cos(z) + 3e-zSin(z) - z - 9
S(z) = the slope of F(z) = (2z + 3e-z)Cos(z) + (3e-z - z2)*Sin(z) - 1

Function F(z) contains periodic functions whose amplitudes increase as z goes to plus and minus infinity. Since F(z) crosses the z-axis an infinite number of times, this equation has an infinite number of real solutions.

To find complex solutions, we'll resolve functions F(z) and S(z) into real and imaginary parts. To accomplish this, we'll substitute the following resolved functions into F(z) and S(z):

z2 = R2 Cos(2q) + iR2 Sin(2q)
Cos(z) = Cos(x)Cosh(y) - iSin(x)Sinh(y)
Sin(z) = Sin(x)Cosh(y) + iCos(x)Sinh(y)
e-z = e-xCos(y) - ie-x Sin(y)
z = x + iy

(you can find these resolved functions in the Table of Resolved Complex Functions). Polar coordinates R and q are defined

R = (x2 + y2)1/2
q = +Arcosine(x/R) when y > 0, and q = -Arccosine(x/R) when y < 0

After simplifying the functions as much as possible, we'll have

Fr = R2Cos(2q)Cos(x)Cosh(y) + R2Sin(2q)Sin(x)Sinh(y) + 3e-xCos(y)Sin(x)Cosh(y) + 3e-xSin(y)Cos(x)Sinh(y) - x - 9

Fi = R2Sin(2q)Cos(x)Cosh(y) - R2Cos(2q)Sin(x)Sinh(y) + e-xCos(y)Cos(x)Sinh(y) - e-xSin(y)Sin(x)Cosh(y) - y

Sr = [2x + 3e-xCos(y)]*Cos(x)Cosh(y) + [2y - 3e-xSin(y)]*Sin(x)Sinh(y) + [3e-xCos(y) - R2Cos(2q)]*Sin(x)Cosh(y) + [3e-xSin(y) - R2Sin(2q)]*Cos(x)Sinh(y)) - 1

Si = [2y - 3e-xSin(y)]*Cos(x)Cosh(y) - [2x + 3e-xCos(y)]*Sin(x)Sinh(y) + [3e-xSin(y) - R2Sin(2q)]*Sin(x)Cosh(y) - [3e-xCos(y) - R2Cos(2q)]*Cos(x)Sinh(y)

As with the real solutions, there are an infinite number of complex solutions.

 

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URL http://members.aceweb.com/patrussell/approximations/MixedEqn.htm
Unpublished Work. © Copyright 2001 Pat Russell. Updated April 17, 2009.