Mathematical Methods for Physics and Engineering : A Comprehensive Guide

3.2 MANIPULATION OF COMPLEX NUMBERS
Find the complex conjugate of the complex numberz=w(3y+2ix),wherew=x+5i. Although we do not ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
Expresszin the formx+iy,when z= 3 − 2 i −1+4i . Multiplying numerator and denominator ...

3.3 POLAR REPRESENTATION OF COMPLEX NUMBERS Rez Imz r θ x y z=reiθ Figure 3.7 The polar representation of a complex number. From ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS Rez Imz r 1 r 2 ei(θ^1 +θ^2 ) r 2 eiθ^2 r 1 eiθ^1 Figure 3.8 The multiplication of two ...

3.4 DE MOIVRE’S THEOREM Rez Imz r 1 r 2 ei(θ^1 −θ^2 ) r 2 eiθ^2 r 1 eiθ^1 Figure 3.9 The division of two complex numbers. As in ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
Expresssin 3θandcos 3θin terms of powers ofcosθandsinθ. Using de Moivre’s theorem, cos ...

3.4 DE MOIVRE’S THEOREM
Find an expression forcos^3 θin terms ofcos 3θandcosθ. Using (3.32), cos^3 θ= 1 23 ( z+ 1 z ) 3 = 1 8 ( ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS Rez Imz e−^2 iπ/^3 e^2 iπ/^3 2 π/ 3 2 π/ 3 1 Figure 3.10 The solutions ofz^3 =1. Not su ...

3.5 COMPLEX LOGARITHMS AND COMPLEX POWERS To avoid the duplication of solutions, we use the fact that−π<argz≤πand find z 1 =2 ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS We may use (3.34) to investigate further the properties of Lnz. We have already noted t ...

3.6 APPLICATIONS TO DIFFERENTIATION AND INTEGRATION 3.6 Applications to differentiation and integration We can use the exponenti ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS 3.7 Hyperbolic functions Thehyperbolic functionsare the complex analogues of the trigon ...

3.7 HYPERBOLIC FUNCTIONS sechx coshx x 1 2 3 4 − 2 − 1 1 2 Figure 3.11 Graphs of coshxand sechx. cosechx cosechx sinhx x 2 4 − 2 ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS cothx cothx tanhx x 2 4 − 2 − 4 − 2 − 1 1 2 Figure 3.13 Graphs of tanhxand cothx. metri ...

3.7 HYPERBOLIC FUNCTIONS 3.7.4 Solving hyperbolic equations When we are presented with a hyperbolic equation to solve, we may pr ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS sech−^1 x sech−^1 x cosh−^1 x cosh−^1 x x 2 4 − 2 − 4 1 2 3 4 Figure 3.14 Graphs of cos ...

3.7 HYPERBOLIC FUNCTIONS sinh−^1 x cosech−^1 x cosech−^1 x x 2 4 − 2 − 4 − 2 − 1 1 2 Figure 3.15 Graphs of sinh−^1 xand cosech−^ ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
Verify the relation(d/dx)coshx=sinhx. Using the definition of coshx, coshx=^12 (ex+e−x ...

3.8 EXERCISES
Evaluate(d/dx)sinh−^1 xusing the logarithmic form of the inverse. From the results of section 3.7.5, d dx ( sinh− ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS (a) the equalities of corresponding angles, and (b) the constant ratio of corresponding ...