For Further Information | HP 15c Scientific specification

Section 11: Calculating With Complex Numbers 137

For Further Information

The HP-15C Advanced Functions Handbook presents more detailed and technical aspects of using complex numbers in various functions with the HP-15C. Applications are included. The topics include:

∙Accuracy considerations.

∙Principal branches of multi-valued functions.

∙Complex contour integrals.

∙Complex potentials.

∙Storing and recalling complex numbers using a matrix.

∙Calculating the nth roots of a complex number.

∙Solving an equation for its complex roots.

∙Using _and fin Complex mode.

Image 137

HP 15c Scientific manual For Further Information

Contents

HP Part Number 00015-90001 Edition 2.4, Sep HP-15C Owner’s Handbook Legal Notice Introduction Contents Display and Continuous Memory Contents Program Branching and Controls Program Editing Subroutines Calculating With Matrices Indirect Display ControlCalculating With Complex Numbers Numerical Integration Appendix C Memory Allocation Contents Appendix a Error ConditionsAppendix D a Detailed Look at Appendix E a Detailed Look at f Function Summary and Index Contents Appendix F BatteriesProgramming Summary and Index Subject Index Quick Look at HP-15C Problem Solver To Compute Keystrokes Display Manual Solutions Keystrokes Display Programmed Solutions 001-42,21,11 KeystrokesDisplay002 003 004 005 006 007 008 009 8313 300.51 HP-15C a Problem Solver Part l HP-15C Fundamentals Getting Started Power On and OffKeyboard Operation Section Changing Signs Prefix KeysKeying in Exponents I O m ´ P I l F T s ? t H b Clear Keys Clears only the last digit Display Clearing ` and −Digit entry not terminated One-Number Functions CalculationsTwo-Number Functions 6532 26.0000 22.0000 5000 17 +13.0000 78.0000 Number Alteration Functions Numeric Functions General Functions One-Number Functions Pressing Calculates Trigonometric OperationsTime and Angle Conversions 7069 Degrees/Radians ConversionsRadians 40.5000 Hyperbolic Functions Logarithmic Functions Two-Number Functions Power FunctionPercentages To Calculate Keystrokes Display Enters the base number the price Polar and Rectangular Coordinate ConversionsCalculates 3% of $15.76 the tax Polar Conversion. Pressing Keystrokes Display Automatic Memory Stack Stack Manipulation Automatic Memory Stack Last X, and Data StorageAutomatic Memory Stack Registers Always displayed Lost Stack Manipulation FunctionsMemory Stack, Last X, and Data Storage Lost 287.0000 Last X Register and K22.2481 12.9000 20.6475 Calculator Functions and the Stack13.9 + +15 X15 Order of Entry and the v Key 7 + Nested Calculations65.0000 69.0000 Arithmetic Calculations With Constants 5 ‛15 Keys 000 Keystrokes Display Growth factor1000 Storing and Recalling Numbers Storage Register Operations322.5000 520.8750 Storage and Recall Arithmetic Clearing Data Storage Registers For recall arithmetic For storage arithmetic Overflow and Underflow Problems24 l-0 15.0000 Memory Stack, Last X, and Data Storage 60.0000 Statistics FunctionsProbability Calculations 270,725.0000 Random Number Generator5764 3422 Registers Accumulating Statistics Register Contents Metric tons per Hectare, y 20.00 40.00 60.00 80.00 Kg per hectare20 z 61v 40 z 7.21 60 z 7.78 80 z l Σy2 20 w 20 z Correcting Accumulated Statistics 40.00 MeanStandard Deviation 31.62 Linear RegressionStandard deviation about the mean nitrogen Application Linear Estimation and Correlation Coefficient Statistics Functions 70 ´j Other Applications Fixed Decimal Display Display Continuous MemoryDisplay Control Engineering Notation Display Scientific Notation Display234568 234567 Special Displays Round-Off ErrorMantissa Display Annunciators Digit Separators Error Display12,345.67 12.345.6700 Status Low-Power IndicationContinuous Memory Resetting Continuous Memory Page Part ll HP-15C Programming Mechanics Programming BasicsCreating a Program Loading a Program ´b a Programming Basics 002 003 004 005 006 007 008 Intermediate Program StopsRunning a Program 300.51 300.51 ´A How to Enter Data Program Memory Totals Radius, r Height, h Base Area Volume Surface Area 004 005 002007-44,40 010 Or G a Instruction Coding Further InformationProgram Instructions Keycode 25 second row, fifth key Memory Configuration 60 ´ m% Initial Memory Configuration ´ m % Program Boundaries19 ´ m% 19.0000 ´bA ´b3 End of memory Unexpected Program StopsAbbreviated Key Sequences Polynomial Expressions and Horners Method User Mode¤ @ y ∕ LOG % 001-42,21,12 Nonprogrammable Functions002 003 004 005 006 007 008 009 0000 12,691.0000 Problems Moving to a Line in Program Memory Program Editing Inserting Program Lines ExamplesDeleting Program Lines Or use Â Single-Step Operations Âhold Line PositionRelease Result Initializing Calculator Status Insertions and Deletions PV 1 + i n Interest+ i n ´bA D ´4 O0 2* O1 2÷ * ´ ´ l0 l1 ´r * n 100 270 Branching Program Branching Controls Test Conditional Tests  n will clear flag number n Flags Example Branching and Looping 013-43,30 010-45,20014 016-44,40 Formula is Example Flags 004-42,21,15 002-43005-43, 4 006-42,21 250.0000 Go to48.0000 10,698.3049 Conditional Branching Looping System Flags Flags 8 Program Branching and Controls Go To Subroutine and Return SubroutinesSubroutine Execution ´b.1 Subroutine Limits 002- R 000 001- ´b9003- O0 004 ´b.5 ´ b.4 Nested Subroutines Subroutine Return 106 Index Register Loop ControlV and % Keys Program Loop Control Indirect Program Control With the Index RegisterIndex Register Storage and Recall Index Register and Loop Control Indirect Branching With Index Register ArithmeticExchanging the X-Register Loop Control With Counters I and e Indirect Flag Control WithIndirect Display Format Control With Start count at zero Count by twos Count up to Nnnnn x x x y y 5 0 0 Iterations Examples Register OperationsStoring and Recalling Keystrokes Display 12.3456 Storage Register Arithmetic Example Loop Control with eExchanging the X-Register −− 011- 42 Loop control number in R2012-42, 5 013- 22 64.8420 0000 50.0000 Example Display Format Control15 O Index Register Contents Indirect Display Control Index Register and Loop Control 118 Part lll HP-15C Advanced Functions Calculating With Complex Numbers Complex Stack and Complex ModeCreating the Complex Stack 120 Entering Complex Numbers Deactivating Complex ModeComplex Numbers and the Stack ´ % hold 8.0000 release Z 8 Y 7 X Keys Manipulating the Real and Imaginary Stacks Stack Lift in Complex Mode Or other operation Clearing a Complex Number − 4 v Continue with any operation Continue with any operation 0000 17.0000 144.0000 Entering Complex Numbers with −. The clearing functions −´ %hold release Followed by another number Entering a Real Number ´ Continue with any operation Entering a Pure Imaginary Number Storing and Recalling Complex Numbers Operations With Complex Numbers´ O L 2 ´ + * ÷ y ¤x N o ∕ @ a 7000 20000428 0491 Complex Results from Real Numbers Polar and Rectangular Coordinate Conversions5708 ´ % hold Release1.5708 Cos θ + i sin θ = re iθ Polar + ib = ∠ θ + 3.1434 84522981 872.0000 352.00002361 4721 For Further Information 138 Calculating With Matrices = A-1B Keystrokes Display Deactivates Complex Mode Running Matrix Dimensions11.2887 2496 Number Rows Columns Dimensioning a Matrix Changing Matrix Dimensions Displaying Matrix Dimensions´mA Keystrokes l B Display Storing and Recalling All Elements in Order Storing and Recalling Matrix Elements ⎡ a Checking and Changing Matrix Elements Individually Keystrokes Display Matrix Descriptors Matrix OperationsStoring a Number in All Elements of a Matrix Result Matrix One-Matrix Operations Copying a Matrix Calculating with Matrices LB b Scalar Operations LA a Elements of Result Matrix LB b 2 LA a 2 Arithmetic OperationsKeystrokes Display Subtracts 1 from the elements Matrix Multiplication Keystrokes Display l a a = AT B Solving the Equation AX = B 2400 24 OA86 OA 8600 274 OB 233 OB 331 OB 120.32 OB 112.96 OB 151.36 OB ´Á Week Cabbage kg 186 141 215 Broccoli kg 116 Calculating the Residual Calculations With Complex Matrices Using Matrices in LU Form Then Z can be represented in the calculator by Storing the Elements of a Complex Matrix Pressing Transforms Into LA a = ⎢ Complex Transformations Between ZP and Z Inverting a Complex Matrix ´ a Multiplying Complex Matrices ´U lC LC lC lC lC lC lC lC ´U Keystrokes lA lB Display Displays descriptor of matrix a ZZ −1 Solving the Complex Equation AX = B AX = B 170.0000 200.0000 1311 03720437 1543 Calculating with Matrices Miscellaneous Operations Involving Matrices Using a Matrix Element With Register OperationsUsing Matrix Descriptors in the Index Register Conditional Tests on Matrix Descriptors Stack Operation for Matrix Calculations Calculating with Matrices Using Matrix Operations in a Program ´m a Summary of Matrix FunctionsKeystrokes Results Calculates residual in result matrix For Further Information 180 UsingFinding the Roots An Equation Clear program memory Finding the Roots of an Equation 001-42,21 ´b0002 003 005 006 007 Desired root Finding the Roots of an Equation ´ bA Keystrokes ¥000 001-42,21,11 003 004 Brings another t-value 5000 1 e tInto X-register 200 t 000 001-42,21 002 003 004 005 When No Root Is Found Error Choosing Initial Estimates Label X + 8 003 004 005 007008 009 6 x + 8 Finding the Roots of an Equation Using in a Program Memory Requirements Restriction on the Use 194 Using fNumerical Integration 002 003 004 1416 7652 4040 Begin subroutine with a label 4401 3825$ ÷ 6054 Accuracy of f ´ i ´ f 7091 8826 382 Using f in a Program Memory Requirements Error 0 Improper Mathematics Operation Error ConditionsAppendix a 205 Error 2 Improper Statistics Operation Error 1 Improper Matrix Operation Error 4 Improper Line Number or Label Call Error 3 Improper Register Number or Matrix ElementError 5 Subroutine Level Too Deep Error 6 Improper Flag Number Pr Error Power Error Digit Entry Termination Stack Lift Last X RegisterStack Lift Appendix B Enabling Operations Disabling Operations Neutral Operations Stack Stack Enabled. disabled 53.1301 No stack LiftAppendix B Stack Lift and the Last X Register Keys Nnn Clear u ¥ \ k + H ∆ \ h ÷ À P* q r c ‘ / N z ∕ P\ o j Last X Register Memory Space Memory AllocationAppendix C Registers Appendix C Memory Allocation M % Function Memory ReallocationMemory Status W 19 ´ m Restrictions on Reallocation´m% 1.0000 Whold 1 64 Automatic Program Memory Reallocation Program Memory Two-Byte Program Instructions Memory Requirements for the Advanced FunctionsIf executed Together Appendix C Memory Allocation How Works Detailed Look atAppendix D 220 Appendix D a Detailed Look at Accuracy of the Root X4 = 000 006 007 008 009 010-43,30 011 012-43,30 013 1718 0681 Interpreting Results´ v B − 45 For 0 x Branch for x ≥ Test for x range3x 45x 2 + End subroutine Initial estimates 000.00001358 Possible root Appendix D a Detailed Look at Bring x-value into X-register ´ b.0 001-42,21,.0 002 003 004 005007 008 009 010 013 014 015 016 10 v ´ ‛ 20 017 018 Finding Several Roots Error 0000 1250 5626 002 003 004 005 006 007 Fx = xx a3 = 6667 Second root Same initial estimatesStores root for deflation Deflated function value Deflation for third root Limiting the Estimation Time Specifying a Tolerance For Advanced InformationCounting Iterations How f Works Detailed Look at fAppendix E 240 X = π1 0π cos4θ − x sinθ dθ Accuracy, Uncertainty, and Calculation Time ´ i ´ f 0000 1416 ´ Clear u Hold Keystrokes Display Return approximation toKeystrokes ´ i Display ´ f ´ Clear u hold 7807 7858 Uncertainty and the Display Format Functions values for example Δx = 0.5×10−n ×10m = aδx dxb = ab 0.5×10−n + m x dx Conditions That Could Cause Incorrect Results ∞ xe− xdx 001-42,21 002- 1 003 004 005 Appendix E a Detailed Look at f Appendix E a Detailed Look at f Conditions That Prolong Calculation Time Keys upper limit into Keys lower limit intoApproximation to integral Uncertainty Appendix E a Detailed Look at f Obtaining the Current Approximation to an Integral For Advanced Information Installing New Batteries Low-Power IndicationBatteries Batteries Appendix F Batteries 2.C 3.H Verifying Proper Operation Self-Tests Complex Functions Function Summary and IndexConversions Digit Entry Index Register Control Display ControlLogarithmic Exponential Functions Mantissa. Pressing 146 MathematicsMatrix Functions To XT Number AlterationTo ZP page164 Probability PercentageStack Manipulation Clear u Storage Statistics Trigonometry 269 Programming Summary and Index Programming Summary and Index 271 Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Modifications Product Regulatory Environment InformationFederal Communications Commission Notice Avis Canadien Canadian Notice Body number is inserted between CE European Union Regulatory Notice