resistor and capacitor in parallel Cole-Cole Plots: Impedance Plots in the Complex **Plane** When we plot the real and imaginary components of impedance in the complex **plane** (**Argand** diagram), we obtain a semicircle or partial semicircle for each parallel RC Voigt/grains, grain boundaries, and electrodes can be seen more distinctly. The complex impedance data can be displayed in the complex impedance **plane** with real part as the abscissa and the imaginary part “ as the ordinate (Cole Cole diagram). A typical /

numbers, so is 2-dimension- al, as was noted by: Caspar Wessel (1745-1818) Jean Robert **Argand** (1768-1822) Carl Friedrich Gauss (1777-1764) 48 The complex **plane** -2 -1 0 1 2 i i -i 2i -2i 2+i 2-i 49 is algebraically closed/. In some sense they arent even related, except, perhaps, historically. 52 Three is better than two? Complex numbers and the complex **plane** proved to be extremely powerful tools for creating precise mathematical models in two dimensions. A natural question: What about a number system for /

on, I shall use H(e -j ) in place of H( ) to emphasise the connection between the -domian and the z-domain. … (7.3) The z-**Plane** & the Unit Circle The frequency response is periodic with period 2 . Thus, we only need evaluate it over (- < < ). Thus, the values of z = e -/2 | arg(z 2 ) z2z2 z 3 = z 2 * We display all of this on a ‘Pole-Zero Plot’. (This is just an **Argand** diagram with a unit circle on and the poles and zeros displayed in the following way). Imag Real X 3 O O O pole zero A pole is /

, b) in a rectangular coordinate system. When a rectangular coordinate system is used to represent complex numbers, the **plane** is called the complex **plane** or the **Argand** **plane**. The x- axis is also called the real axis, because the real part of a complex number is plotted/Slide 7.5 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Plotting Complex Nubmers Plot each number in the complex **plane**. A: 1 + 3i B: –2 + 2i C: –3 D: –2i E: 3 – i Solution Slide 7.5 - 7 Copyright ©/

past any unknown object can be represented as a complex potential. In particular we define the complex potential In the complex (**Argand**-Gauss) **plane** every point is associated with a complex number In general we can then write Now, if the function f is analytic, /that it is also differentiable, meaning that the limit so that the derivative of the complex potential W in the complex z **plane** gives the complex conjugate of the velocity. Thus, knowledge of the complex potential as a complex function of z leads to /

Impedance Vector Impedance ‘resistance’ Admittance ‘conductance’: Representation of impedance value, Z = a +jb, in the complex **plane** (see also http://math.tutorvista.com/number-system/absolute-value-of-a-complex-number.html Data Presentation: Nyquist Plot with Impedance / Nyquist Plot with Impedance Vector When we plot the real and imaginary components of impedance in the complex **plane** (**Argand** diagram), we obtain a semicircle or partial semicircle for each parallel RC Voigt network: Nyquist plot or also/

1 ) cos( 2 ) exp(-i 1 ) A ┴ = sin( 1 ) sin( 2 ) exp(-i 2 ) Transversity **plane** in the J/ rest frame B meson defines positive x-axis K (KK) **plane** defines (x,y) **plane** K (K + ) +ve y momentum T, T polar & azimuthal angles of + K helicity angle of K* ( ) 15 /-0.09±0.13±0.06 This measurem.0.63 ±0.04 ±0.030.18 ±0.05 ±0.053.07 ±0.40 ±0.070.11±0.23±0.06 17 **Argand** Diagrams dominant longitudinal polarization L / = |A 0 | 2 non-zero parity-odd fraction ┴ = |A ┴ | 2, (P-wave, CP= -1 /

= 2 r. ( s - s 0 ) = 2 r.S | s 0 |=| s |=1/ |S|=2sinθ/λ Diffraction Scattering by a crystal – Braggs law: diffraction as reflection from crystal **planes** d path difference = 2dsin θ For constructive interference, nλ = 2dsinθ | S | = 2sin / = 1/d If a diffraction pattern fades out at an angle of 2 / vectors are pointing in different directions. t=0 t=1 2 i t a. S t=2 t=3 t=4 t=5 t=6 **Argand** diagramm Laue Conditions: a.S = h b.S = k c.S = l h, k, l are whole numbers either zero, positive or /

form: Addition: Copyright R. Janow – Fall 2015 Representation using the Complex **Plane** Imaginary axis (y) Real axis (x) x y |z| Magnitude 2 : Polar form: Argument: “**Argand** diagram” Picture also displays complex-valued functions Copyright R. Janow – Fall 2015/ Re Let a(t) be a sinusoidally varying real quantity to be pictured as a vector in the complex **plane** rotating (counterclockwise) at angular frequency Copyright R. Janow – Fall 2015 Complex exponentials (time domain phasors) are solutions of /

, isolated resonance For broad, overlapping and many channel resonances we need a more general approach K-matrix formalism (**Argand** diagram) Int. WE Heraeus Summer School, September 2 nd, 2005 F. Martínez-Vidal, Measurements of the CKM/shapes of selection variables (mES, DE, Fisher, etc.) –Background fractions and Dalitz shapes –Efficiency variations across Dalitz **plane** (including tracking efficiency) –Invariant mass resolution –Biases from control samples Dalitz model systematic uncertainty includes: –No /

amplitude, not changing the values of differential cossection and asymmetry while affecting A and R. In the w-**plane** such transformation is equivalent to mirroring of a trajectory across the unit circle crossing points are critical for branching / constatnt or linearly varying background M resonance mass full width R= EL / 2 resonance circle radius on **Argand** plot pole residue phase B background parameters Pole parameters for all 7 -isobars in the second resonance region were /

PART Copyright R. Janow – Fall 2015 Representation using the Complex **Plane** Imaginary axis (y) Real axis (x) x y |z| Magnitude 2 : Polar form: Argument: “**Argand** diagram” Picture also displays complex-valued functions Copyright R. Janow –/” A (often simply called a “phasor”): Represent sinusoidally varying real quantity a(t) as a vector in the complex **plane**, rotating (counterclockwise) at angular frequency Copyright R. Janow – Fall 2015 Phasor Definitions, continued A phasor represents a sinusoidal/

Flow field can be represented as a complex potential. In particular we define the complex potential In the complex (**Argand**-Gauss) **plane** every point is associated with a complex number In general we can then write The Cauchy-Riemann conditions are: Differentiating/as a complex function of z leads to the velocity field through a simple derivative. Fertile Complex Potentials ELEMENTARY IRROTATIONAL **PLANE** FLOWS The uniform flow The source and the sink The vortex Creation of mass & Momentum in Space by Nature THE/

the sum of a real and imaginary part maintains a good algebraic field. For example, Complex Numbers as Vectors Jean-Robert **Argand** in 1806 came up with the idea of a geometrical interpretation of complex numbers. Replace the x-axis with the real / the vector, and e iθ gives the direction. Physics Application: Centripetal Motion Create a parametrization of the position in the complex **plane**, Consider an object moving in a circle of radius r with an angular frequency of ω. What is its velocity and acceleration?/

one of which real numbers are represented, and on the other pure imaginaries, thus providing a reference for graphing complex numbers. **Argand** diagram E. Some unusual mathematical terms Are We Speaking the Same Language? Perfect number: A number equal to the sum of all/? Quadrinomial: a four-term polynomial. I propose the following new definitions: Axial points: Points on an axis. In the x-y-**plane**, an axial point is any point that has at least one 0 coordinate. Betweenness inequality: a < x < b. This means /

Phase probability distribution Radii are F p and k*F ph Width are p and k* ph The red area are the places in **Argand** space where both F P and F PH -F H can be Most probable versus best phase The most probable phase is not necessarily the “best/ centric, maving phase = 0° or 180° If the crystal has 2-fold, 4-fold or 6-fold rotational symmetry, then the reflections in the 0-**plane** are centric. (Because the projection of the density is centrosymmetric) For centric reflections: |F ph | = |F p | ± |F h | This /

general The conjugate Complex Conjugate Phasor = Rotating Arrow + Associated Phase AnglePhasor Representation of a complex number in terms of real and imaginary components Im Complex **Plane** r sin r cos Re z ^ **Argand**/Phasor Diagram © SPK Complex **Plane** Taylor series is a series expansion of a function about a pointseries expansionfunction 1-d Taylor series expansion of a real function about a point/

general The conjugate Complex Conjugate Phasor = Rotating Arrow + Associated Phase AnglePhasor Representation of a complex number in terms of real and imaginary components Im Complex **Plane** r sin r cos Re z ^ **Argand**/Phasor Diagram © SPK Complex **Plane** Consider the following Maclaurin series expansions Expansions are valid for complex arguments x too Function : Complex Numbers & Simple Harmonic Oscillations March 19, 2016March 19, 2016March/

the vector by -1 and it points the other way i.e. a 180° shift As -1 = j2 then j lies between them i.e. a 90° shift Complex **plane** or **argand** diagram j -c+jd Imaginary a+jb j2= -1 Real g-jh -e-jf j3=-j Polar form of a complex number The number may be represented by its vector/

(rationalizing the denominator) For some reason we dont like is in the denominator. So we rewrite the fraction by multiplying by the complex conjugate. Example Plotting Complex Numbers (on the **Argand** **plane**) i R -5 – 2 i 2 – 2 i 2 + 3 i -5 -2 + 3 i -3 i

DIVISOR divided by itself z2 z1 = (x22 + y22) z2 z2* z1z2* x1 x2 + y1 y2 + j y1 x2 – x1 y2 Division Complex No.s – Graphically The **Argand** diagram Modulus (magnitude) of z r = mod z = |z| = x2 + y2 x y r z = x + i y Im(z) Re(z) Argument (/ Euler Identity Where Then from the Vector Plot Need to be careful abaout the values of θ when z is in the LEFT-half of the **Argand** **Plane** => must Add 180 degrees Complex Number Calcs cont Consider Two Complex Numbers The PRODUCT n•m The SUM, Σ, and DIFFERENCE, , for these/

+ b22)(a32 + b32) (d) (a12 + a22 + a32)(b12 + b22 + b32) Choose the Correct Answer 7. The points z1, z2, z3, z4 in the complex **plane** are the vertices of a parallelogram in order if and only if (a) z1+ z4 = z2 + z3 (b) z1+ z3 = z2 + z4 (c) z1+ z2 = /z3 + z3 (d) z1 – z2 = z3 – z4 8. If a = 3 + i and z = 2 – i, then the points on the **Argand** diagram representing az, 3az and – az are (a) vertices of a right angled triangle (b) vertices of an equilateral triangle (c) vertices of an isosceles triangle (d) /

spectator model: Orthogonality relation: γd→π 0 np Amplitude: dσ/dΩ suppressed at θ π → 0 Orthogonality is ignored is a **plane** wave andare eigenstates of the same Hamiltonian 15 200 Absorption of pions 400600800 Photon energy [MeV] 0 2 4 6 x10 2 σ [μb] σ/the resonance region ΔQ ~ 2/Γ R strong influence of inelastic channels Example: ηd elastic scattering in the S 11 (1535) region **Argand** diagram for L=0Inelasticity parameter Three-body calculation S 11 (1535) Complex nuclei: σ(γA → ηX) ~ A 2/3 (surface /

) | z | 2 (c) 2| z |(d) 2| z | 2 29.If a = 3 + i and z = 2 – 3i then the points on the **Argand** diagram representing az, 3az and – az are (a) vertices of a right triangle (b) vertices of an equilateral triangle (c) vertices of an isosceles triangle (d) collinear Choose / and is (a) (2, 1, 1) (b) (1, 2, 1) (c) (1, 1, 2) (d) (1, 1, 1) 34.The distance from the origin to the **plane** is (a) (b) (c) (d) Choose the Correct Answer 35.If,, are three mutually perpendicular unit vectors, then | + + | = (a) 3 (b) 9 (c) 3 /

map shows a series of condition vectors derived from 20 tests. This map is a visual representation of the condition of the circuit breaker represented as a point in an **Argand** **plane**. METHODOLOGY The Wavelet Packet Transform is originally applied to the fault diagnosis of circuit breakers and the algorithm is tested using the real data collected. The wavelet maxima are used/

of these numbers you get 1. (Try it and see!) We found the cube roots of 1 were: Lets plot these on the complex **plane** about 0.9 Notice each of the complex roots has the same magnitude. Also the three points are evenly spaced on a circle. This will /always be true of complex roots. each line is 1/2 unit This representation known as **Argand** diagram Steps to Find Roots of Complex Numbers 1)Change complex number to polar form z = r cis θ 2)Find the nth roots: 3/

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Complex No.s – Graphically x y r z = x + i y Im(z) Re(z) The **Argand** diagram Modulus (magnitude) of z arctan = arg z = x y + , if x < 0 arctan xy, if x > 0 r = mod z =//Physics 25: Computational Methods Complex Number Calcs Consider a General Complex Number This Can Be thought of as a VECTOR in the Complex **Plane** This Vector Can be Expressed in Polar (exponential) Form Thru the Euler Identity Where Then from the Vector Plot x y r/

Unitarity Triangle Three complex numbers, which sum to zero Divide by so that the middle element is 1 (and real) Plot as vectors on an **Argand** diagram If all numbers real – triangle has no area – No CP violation Real Imaginary Hence, get a triangle ‘Unitarity’ or ‘CKM triangle’ /Triangle if SM is correct. Otherwise triangle will not close, Angles won’t add to 180 o 23 Unitarity conditions hence 6 triangles in complex **plane** j=1,3 No phase info. j,k =1,3 j k db: sb: ds: ut: ct: uc: 24 CKM Triangle -/

many waves Complex method of Addition Resultant of N waves Complex amplitude of the resultant wave For N=2 Phasor Addition of Waves **Argand** Diagram Representation of a complex number in terms of real and imaginary components In Complex **plane** A Sin t A Cos wt Re Rotating Arrow + Associated Phase Angle = Phasor Addition of two phasor Resultant Addition of three phasor/

time set the derivative to zero 35 Principle of superposition 35 36 Principle of superposition 37 38 Resultant phasor Phasor Addition of waves **Argand** Diagram Representation of a complex number in terms of real and imaginary components In Complex **plane** A Sin t A Cos wt Re Rotating Arrow + Associated Phase Angle = Phasor Addition of two phasor Resultant Addition of three phasor/

CAS antitrust compliance policy. 3 Outline I.Complex Numbers II.Roots-of-Unity Random Variables III.Collective-Risk Model IV. Examples 4 I. Complex Numbers Basic Arithmetic Complex **Plane** versus Real Line Wessel(1979)-**Argand**(1806) diagram [Kramer, 73] No trichotomy between complex numbers 5 I. Complex Numbers (Cont’d) Completeness of Complex Numbers Fundamental Theorem of Algebra [Kramer, 71] Actuaries should/

is the modulus of z, orIf is the modulus and is always positive or If what is is the complex conjugate of z. Graphing Complex Numbers use the **Argand** Diagram or the Complex **Plane**. x-axis would be the real numbers y-axis would be the imaginary numbers Graph the complex number z = 2 + 3i Graph the complex number z = -4i If/

can be analyzed with reference to a transformed problem in which: Associated eigenvalues depends only on J c (f) and J f (c) The eingenvalues have to be (in the **Argand**-**plane** of the imaginary numbers) within an elipse that only depends ○ On α and β in absence of ATIS ○ On α, β and max in presence of (accurate) ATIS The effect of max is to/

Gershon, IoP Masterclass, March 18 th 2009 The Unitarity Triangle ● Squares of CKM matrix elements describe probabilities ⇒ matrix must be unitary 12 Three complex numbers add to zero ⇒ triangle in **Argand** **plane** Tim Gershon, IoP Masterclass, March 18 th 2009 The BABAR Experiment 13 ● PEP-II accelerator collides electrons and positrons at energies tuned to produce pairs of B mesons ● Electron/positron/

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