Chapter Calculations with complex numbers, Definitions | HP 49g

Chapter 4

Calculations with complex numbers

This chapter shows examples of calculations and application of functions to complex numbers.

Definitions

A complex number z is written as z = x + iy, (Cartesian form) where x and y are real numbers, and i is the imaginary unit defined by i2 = -1. The number has a real part, x = Re(z), and an imaginary part, y = Im(z). The polar form of a complex number is z = re iθ = r⋅cosθ + i r⋅sinθ, where r = z

= x 2 + y 2 is the modulus of the complex number z, and θ = Arg(z) =

arctan(y/x) is the argument of the complex number z. The complex conjugate of a complex number z = x + iy = re iθ, isz = x – iy = re -iθ . The negative of z, –z = -x-iy = - re iθ, can be thought of as the reflection of z about the origin.

Setting the calculator to COMPLEX mode

To work with complex numbers select the CAS complex mode:

H@@CAS@) ˜˜™@@CHK

The COMPLEX mode will be selected if the CAS MODES screen shows the option _Complex checked off, i.e.,

Press @@OK@@ , twice, to return to the stack.

Page 4-1

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HP 49g manual Chapter Calculations with complex numbers, Definitions, Setting the calculator to Complex mode

Contents

User’s manual Hp 49g+ graphing calculator Printing History Preface Table of Contents Calculations with real numbers Algebraic and arithmetic operations Calculations with complex numbers Operations with lists Solution to equations Matrices and linear algebra Vectors Vector Analysis Applications GraphicsCalculus Applications Multi-variate Calculus Applications Statistical Applications Differential EquationsProbability Distributions Numbers in Different Bases Using SD cardsLimited Warranty W-1 Batteries Chapter Getting startedBasic Operations Contents of the calculator’s display Tool menu Menus Introducing the calculator’s keyboard Setting time and datePage Selecting calculator modes Operating Mode √ 3.*5.-1/3.*3./23.3+EXP2.5 Lets try now the expression proposed earlier Number Format and decimal dot or comma Page Page Page Angle Measure Coordinate System Selecting CAS settings Selecting Display modes Explanation of CAS settings Selecting the display font Selecting properties of the line editor Selecting properties of the Stack Selecting properties of the equation writer EQW References Creating arithmetic expressions Chapter Introducing the calculatorCalculator objects Editing expressions in the stackPage Page 2L 1 + + 2 L + y Creating algebraic expressions Using the Equation Writer EQW to create expressions Page + 2 ⋅ 5 + + e  x + 2µ ⋅ ∆y  Subdirectories Organizing data in the calculatorHome directory Typing variable names Variables Creating variables Page Page Checking variables contents Page Deleting variables Using function Purge in the stack in Algebraic mode Choose boxes vs. Soft Menu Using function Purge in the stack in RPN modeUndo and CMD functions Show Prog menu list and select Memory Page References Examples of real number calculations Chapter Calculations with real numbersPage Page Using powers of 10 in entering data Page Hyperbolic functions and their inverses Real number functions in the MTH menuPage Units menu Operations with unitsPage Available units Attaching units to numbers Operations with units Page Unit conversions Physical constants in the calculator Engl Defining and using functions ‘LNx+1 + EXPx ‘ Reference Definitions Setting the calculator to Complex modeChapter Calculations with complex numbers Polar representation of a complex number Entering complex numbers Simple operations with complex numbers Cmplx menu through the MTH menu Cmplx menus Cmplx menu in keyboard Function Droite equation of a straight line Functions applied to complex numbers Reference Entering algebraic objects Chapter Algebraic and arithmetic operations Simple operations with algebraic objects `‚¹ Collect Functions in the ALG menuPage Operations with transcendental functions Expansion and factoring using log-exp functionsExpansion and factoring using trigonometric functions Functions in the Arithmetic menu Horner function Polynomials Quot and Remainder functions VariablePcoef function Proot function SIMP2 function FractionsPeval function Fcoef function Propfrac functionPartfrac function − 5X 2 + 3X − Step-by-step operations with polynomials and fractionsFroots function Reference Function Isol Chapter Solution to equationsSymbolic solution of algebraic equations Function Solve Page Function Zeros Function Solvevx Numerical solver menu @SOLVE@ Polynomial Equations Solve for coefficients ˜@SYMB@ Solving equations with one unknown through NUM.SLV Financial calculations Solution to simultaneous equations with Mslv Page Reference Changing sign Chapter Operations with listsCreating and storing lists Operations with lists of numbers Addition, subtraction, multiplication, division Functions applied to lists MTH/LIST menu Lists of complex numbersLists of algebraic objects ∆LIST MAP function SEQ function Typing vectors in the stack Chapter VectorsEntering vectors Storing vectors into variables in the stack Using the Matrix Writer Mtrw to enter vectorsPage Page Addition, subtraction Simple operations with vectors Absolute value function Multiplication by a scalar, and division by a scalar Dot product MTH/VECTOR menuMagnitude Cross product Entering matrices in the stack Using the Matrix WriterChapter Matrices and linear algebra Typing in the matrix directly into the stack Addition and subtraction Operations with matrices Multiplication Cij = ∑aik ⋅ bkj , for i = 1,2,K, m j = 1,2,K, n Page Function Trace Solution of linear systemsCharacterizing a matrix The matrix Norm menu Function DET  M Using the numerical solver for linear systems=  a x2 Solution by division of matrices Solution with the inverse matrix Graphs options in the calculator Chapter Graphics = 1 exp− x Plotting an expression of the form y = fx Generating a table of values for a function Page Fast 3D plots Left-1 Right1 Near-1 Far Low High Step Indep Depnd Page Reference Function lim Chapter Calculus ApplicationsCalc Calculus menu Limits and derivatives Functions Deriv and Dervx Functions INT, INTVX, RISCH, Sigma and Sigmavx Anti-derivatives and integralsXdx = F x + C Definite integrals Infinite series Functions TAYLR, TAYLR0, and Series Reference Partial derivatives Chapter Multi-variate Calculus Applications Φ x, ydydx Multiple integrals ∇ = i ⋅ ∂ + j ⋅ ∂ + k ⋅ ∂ ∂x ∂y ∂z Chapter Vector Analysis ApplicationsDel operator Gradient Curl Divergence CALC/DIFF menu Solution to linear and non-linear equationsChapter Differential Equations Function Ldec Yx = 5 ⋅ exp−x3 / 3 ⋅ expx3 / 3 ⋅ dx + C0 Function Desolve Desolve Variable Odetype Laplace transform and inverses in the calculator Laplace Transforms Fourier series for a quadratic function Fourier series= ∑ c n Function FourierPage Reference − 2... − + 1 = − r Chapter Probability DistributionsMTH/PROBABILITY.. sub-menu part Factorials, combinations, and permutations Random numbers Student-t distribution MTH/PROB menu partNormal distribution F distribution Chi-square distribution Calculating single-variable statistics Chapter Statistical ApplicationsEntering data Page Obtaining frequency distributions Fitting data to a function y = fx 195238095238 + 2.00857242857*X Correlation Covariance Obtaining additional summary statistics Confidence intervals Page Hypothesis testing Page Reference Writing non-decimal numbers Chapter Numbers in Different BasesBase menu HEX DEC OCT BIN Storing objects in the SD card Using SD cardsChapter Purging an object from the SD card Recalling an object from the SD card Limited Warranty Service HP Invent USA Regulatory information