Microsoft COM for Complex Variables
Microsoft COM for Complex Variables
Microsoft COM for Complex Variables
Dr. G. B. Swartz
Professor Mathematics
Monmouth University
W. Long Branch, NJ 07764
[email protected]
ICTCM
The Twelfth Annual International Conference on
Technology in Collegiate Mathematics
Burlingame, CA
November 5, 1999
Abstract
Our senior mathematics course in Complex Variables uses
technology for complex arithmetic and mappings. Microsoft COM
technology is used to produce a component in Visual Basic code that
can be used in programs or in Excel. The component encapsulates the
data structure and operations for complex arithmetic. The component
is used in programming projects for complex valued computations and
plotting. We show example projects and how to create the COM
complex component.
Introduction
Complex numbers are composed of two real quantities.
Arithmetical operations on such manipulate the two and produce
results using real arithmetic. We usually program this one quantity
at a time. However in a course focused on Complex Variables this is
unproductive work for the participants. We create for them a
component that allows reference to a complex number and its
arithmetic by normal programming constructions. The technology that
allows us to do this with ease is the Microsoft COM technology for
encapsulating a data structure and its operations in a programming
unit that can be referenced in a program. The component is
connected to an Excel spreadsheet using Visual Basic Application
(VBA is the macro language built into Excel) to compute the complex
values. The class project included: the computation of parallel
impedances, plotting the complex valued output of a bandpass
filter, using complex valued functions to estimate the real
derivative of a function, to make a 3d plot of the magnitude of a
complex valued function of a complex variable, and finally to show
how a circle in the complex plane is mapped by a complex valued
function of a complex variable.
Components are written in Visual Basic 6, tested and compiled as
an active X dll. The extension of the compiled file is that of a
dynamic linked library, a dll. Components have the advantage that
they are protected from the user so that a user cannot modify the
code and they provide a natural syntax to the inherently
complicated syntax of complex arithmetic.
The Complex Com
The complex com is written to support the following complex
operations. These are grouped as properties and methods, which are
classified as subroutines or functions.
Under properties we have Re and Im for the real and imaginary
part of the complex number. Under methods of the type subroutines
we have: Create( real, imag), Recip() for the reciprocal of the
current complex number, Add(z) and Mult(z) for addition of z or
multiplication of z by the current complex number. Under functions
we have: Magnitude() for the absolute value and
FormatComplex(Optional n=4) which returns a string formatted with n
decimal places of the Cartesian format for a complex number.
At the end of this paper we will extend the Com to include a
dependent Com, Circle, to show how that is done.
Basic Complex Arithmetic
We illustrate the basic of complex arithmetic and show how it
would be done using ComComplex. Let z1=a+bi and z2=c+di. The x term
is called the real part of z, in Com it is z1.Re; and y is the
imaginary part, z1.Im.The sum is a+c + (b+d)i, in Com it would be
done as z1.Add z2 and z1 would contain the sum. Multiplication
results in ac-bd + (bc+ad)i, in Com as z1.Mult z2. For example
(3+4i).Mult (1-2i) results in 11-2i. The magnitude is
|z|=sqrt(x2+y2) and in com as a=z.magnitude. The format command
makes z1 into a string for printing, z1.FormatComplex() results in
3.00 + I(4.00). The Create command makes the object, thus to
establish z1 in code we issue the command:
z1.Create(real:=3,imag:=4), using named arguments.
Using Excel
To use these commands on an Excel spreadsheet we need to use VBA
to take a cell value and send it to the com and then use VBA to
send the result back to the cell. To have Excel multiply two
complex numbers, z1 and z2, you enter 3 and 4 into cells A1 and B1,
enter 1 and 2 into C1 and D1. You select the next two cells, E1 and
F1, and enter the function =Multiply(a1,b1,c1,d1) and array enter
the function. Array enter means to press CTRL+SHIFT+ENTER. The
function in VBA calls Com and returns the answer, 11 and 2 to the
two cells. In the VBA code, which you get to by selecting
Tools/Macro/Visual Basic Editor, you enter:
Function Multiply(a,b,c,d)
Dim z as Complex, u as Complex
Dim P(1,1)
Set z = New Complex
z.Create a, b
Set u = New Complex
u.Create c, d
z.Mult u
P(0,0)=z.Re
P(0,1)=z.Im
Multiply=P
End Function
The Dim commands identify the Com objects and the array P, and
the Set command instantiates each object. To return values to the
Excel spreadsheet we must use an array, P, and assign it to the
name of the function, as in the last line. To connect the
Complex.dll file to the program you select in the VB Editor the
menu: Tools/Reference/Browse and click on the file complex.dll.
Parallel Impedance Project
The first project is to find the parallel impedance Z of a
circuit with impedance Z1 in parallel with a circuit of impedance
Z2. The parallel impedance Z=1/(1/Z1+1/Z2) is a fairly difficult
complex calculation. We did this with a resonant circuit of an
inductor with Z1=iwL and Z2=i/wC. This has infinite Z at
wo=1/sqrt(LC), as is well known. If resistance is added to L, as it
would be due to coil resistance, the point of max|Z| is the
solution of a quintic in w and is not a simple formula, we found
using Maple that the point of resonance is given approximately by
the following formula, this corrects a published formula in
Boylestad, Introductory Circuit Analysis, 7ed,Merrill, 1994. We
found w0adj=wo*sqrt(1-(R2C/L)2/2). We used Maple to investigate
this in a symbolic manner.
Maple code and approximate resonant frequency for R+iwL in
parallel with i/(wC):
> Z1:=A+I*w*L; Z2:=B+I/(w*C);
> Z:=simplify(evalc(1/(1/Z1+1/Z2)));
> mag:=simplify(evalc(abs(Z)^2)):
> dmag:=simplify(diff(mag,w)):
> sol:=solve(dmag=0,w):
> w1:=sol[5];
This led us to investigate computation with a balanced circuit
of R in both the inductor and capacitor. We found, what is a new
result, that the resonance does not detune. The project plots the
selectivey as |Z|/Zmax, where Zmaz=R(1+L/R2C)2/2. The VBA code
is:
Function parallel(a, b, c, d)
Dim p(1, 2)
Dim z1 As New Complex
Dim z2 As New Complex
z1.Create a, b
z2.Create c, d
z1.recip
z2.recip
z1.Add z2
z1.recip
p(0, 0) = z1.Re
p(0, 1) = z1.Im
parallel = p
End Function
The worksheet output is in Table 1, and the charts are in Figs 1
and 2.
Table 1
w
A
L
B
C
Z
w
Mag
0.200
0.200
0.400
0.200
-20.000
0.2083
0.4060
0.200
0.4563
A=1
0.400
0.200
0.800
0.200
-10.000
0.2375
0.8636
0.400
0.8957
L=2w
0.600
0.200
1.200
0.200
-6.667
0.3060
1.4483
0.600
1.4803
0.800
0.200
1.600
0.200
-5.000
0.4717
2.3092
0.800
2.3569
B=1
1.000
0.200
2.000
0.200
-4.000
0.9654
3.8269
1.000
3.9468
C=-4/w
1.200
0.200
2.400
0.200
-3.333
3.2879
7.2052
1.200
7.9199
1.400
0.200
2.800
0.200
-2.857
19.7020
2.7860
1.400
19.8980
1.600
0.200
3.200
0.200
-2.500
5.0985
-8.5723
1.600
9.9739
1.800
0.200
3.600
0.200
-2.222
1.7469
-5.3283
1.800
5.6074
2.000
0.200
4.000
0.200
-2.000
0.9654
-3.8269
2.000
3.9468
2.200
0.200
4.400
0.200
-1.818
0.6665
-3.0108
2.200
3.0837
2.400
0.200
4.800
0.200
-1.667
0.5191
-2.4997
2.400
2.5530
2.600
0.200
5.200
0.200
-1.538
0.4347
-2.1483
2.600
2.1918
2.800
0.200
5.600
0.200
-1.429
0.3813
-1.8908
2.800
1.9289
3.000
0.200
6.000
0.200
-1.333
0.3451
-1.6933
3.000
1.7281
3.200
0.200
6.400
0.200
-1.250
0.3193
-1.5364
3.200
1.5692
3.400
0.200
6.800
0.200
-1.176
0.3002
-1.4084
3.400
1.4400
3.600
0.200
7.200
0.200
-1.111
0.2855
-1.3017
3.600
1.3326
3.800
0.200
7.600
0.200
-1.053
0.2740
-1.2112
3.800
1.2418
4.000
0.200
8.000
0.200
-1.000
0.2648
-1.1334
4.000
1.1640
4.200
0.200
8.400
0.200
-0.952
0.2572
-1.0657
4.200
1.0963
4.400
0.200
8.800
0.200
-0.909
0.2510
-1.0062
4.400
1.0370
4.600
0.200
9.200
0.200
-0.870
0.2458
-0.9533
4.600
0.9845
4.800
0.200
9.600
0.200
-0.833
0.2413
-0.9061
4.800
0.9377
5.000
0.200
10.000
0.200
-0.800
0.2375
-0.8636
5.000
0.8957
5.200
0.200
10.400
0.200
-0.769
0.2343
-0.8251
5.200
0.8577
5.400
0.200
10.800
0.200
-0.741
0.2314
-0.7901
5.400
0.8233
5.600
0.200
11.200
0.200
-0.714
0.2289
-0.7580
5.600
0.7918
5.800
0.200
11.600
0.200
-0.690
0.2267
-0.7286
5.800
0.7631
6.000
0.200
12.000
0.200
-0.667
0.2248
-0.7015
6.000
0.7366
6.200
0.200
12.400
0.200
-0.645
0.2230
-0.6764
6.200
0.7122
6.400
0.200
12.800
0.200
-0.625
0.2215
-0.6531
6.400
0.6896
6.600
0.200
13.200
0.200
-0.606
0.2201
-0.6314
6.600
0.6687
6.800
0.200
13.600
0.200
-0.588
0.2188
-0.6112
6.800
0.6492
7.000
0.200
14.000
0.200
-0.571
0.2176
-0.5922
7.000
0.6310
7.200
0.200
14.400
0.200
-0.556
0.2166
-0.5745
7.200
0.6140
7.400
0.200
14.800
0.200
-0.541
0.2156
-0.5578
7.400
0.5980
7.600
0.200
15.200
0.200
-0.526
0.2148
-0.5421
7.600
0.5831
7.800
0.200
15.600
0.200
-0.513
0.2140
-0.5272
7.800
0.5690
8.000
0.200
16.000
0.200
-0.500
0.2132
-0.5132
8.000
0.5557
8.200
0.200
16.400
0.200
-0.488
0.2126
-0.4999
8.200
0.5432
8.400
0.200
16.800
0.200
-0.476
0.2119
-0.4873
8.400
0.5314
8.600
0.200
17.200
0.200
-0.465
0.2114
-0.4754
8.600
0.5203
8.800
0.200
17.600
0.200
-0.455
0.2108
-0.4640
8.800
0.5097
9.000
0.200
18.000
0.200
-0.444
0.2103
-0.4532
9.000
0.4996
Fig. 1
Z vs w Z1=1+i2w, Z2=1-4i/w
Resonance at 1.414
0.0000
5.0000
10.0000
15.0000
20.0000
25.0000
0.000
2.000
4.000
6.000
8.000
10.000
Fig. 2
|Z|/max vs w
Z1=R+i2w, Z2=R-4i/w
Resonance at 1.414, R=1, 0.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0.5
1
1.5
2
2.5
3
1
h
m
0.150
-0.49994
0.100
-0.49999
0.050
-0.50000
0.025
-0.50000
0.010
-0.50000
BandPass Filter Project
The second project is to plot the complex valued output of the
transfer function for a low pass filter, LP, a high pass filter, HP
and their product, a bandpass filter, BP. Most engineers are
accustomed to think of these plots in terms of magnitude where they
show a gain of 1 over the pass region of frequencies and near 0
outside the pass range. However, the output is a complex valued
quantity, so in this project we plot the Cartesian complex number
of the output as frequency varies. The code in the worksheet
is:
Function BandPass(f, q)
Dim p(1, 2)
Dim z As Complex
Set z = New Complex
Dim u As Complex
Set u = New Complex
f1 = -1 / f: i2 = q * f
z.Create real:=1, imag:=f1
u.Create real:=1, imag:=i2
z.recip
u.recip
z.mult u
p(0, 0) = (1 + q) * z.Re
p(0, 1) = (1 + q) * z.Im
BandPass = p
End Function
The chart is:
BandPass with parameter Frequency
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.00
0.20
0.40
0.60
0.80
1.00
1.20
Estimate of Real Derivative
The real derivative of a real valued function, f(x), can be
obtained from IM(f(x+ih)/h) for a small value of h. This estimate
is better than the usual (f(x+h)-f(x))/h or (f(x+h)-f(x-h))/2h. The
VBA code is:
Function diff(a, h)
Dim z As Complex
Dim w As Complex
Set z = New Complex
z.Create real:=a, imag:=h
Set w = Witch(z)
diff = w.Im / h
End Function
The function diff call a function Witch(z) which is the Witch of
Agnassi, 1/(1+z2). The code for it illustrates how to write a
function that returns an object, in this case a complex object:
Function Witch(z As Complex) As Complex
Dim w As Complex
Set w = New Complex
z.mult z
kk = z.Re + 1
jj = z.Im
w.Create real:=kk, imag:=jj
w.recip
Set Witch = w
End Function
The result is shown in the chart showing the tangent line at
x=1:
0.4300
0.4500
0.4700
0.4900
0.5100
0.5300
0.5500
0.5700
0.850
0.950
1.050
1.150
The worksheet is set up in terms of h for the value of x in cell
A1, the function is called as =diff(A3,$A$1). The worksheet is in
Fig 3.
Fig. 3.
|Z|/max vs w
Z1=R+i2w, Z2=R-4i/w
Resonance at 1.414, R=1, 0.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0.5
1
1.5
2
2.5
3
3D Plot of A Complex Valued Function
Visualizing a map of a complex valued function of a complex
variable is not easy due to the high dimension of the graph space.
We can, however, show the magnitude of the function over the
complex plane in a 3D plot. This project shows the magnitude of the
Witch function on the plane defined by [0..3] X [0..2].
0.5
1.5
2.5
3.5
0.5
3
0
0.2
0.4
0.6
0.8
1
The worksheet is set up as an array of the real and imaginary
parts of the domain:
1
h
m
0.150
-0.49994
0.100
-0.49999
0.050
-0.50000
0.025
-0.50000
0.010
-0.50000
The code in VBA in the Excel project for the worksheet is:
Function Witch2(a, b)
Dim w As Complex
Set w = New Complex
Dim z As Complex
Set z = New Complex
z.Create real:=a, imag:=b
z.mult z
kk = z.Re + 1
jj = z.Im
w.Create real:=kk, imag:=jj
w.recip
Witch2 = w.magnitude
End Function
How to Create a COM Object
The process to make the COM object is to write a class in Visual
Basic 6, VB6, and compile it as a DLL. Start VB6 with an Active X
dll project. Add a class module to the project. Make sure its
instancing property is set to 5-Multiuse on the property sheet for
the class. Make the name property of the class: complex. The
properties of a class are viewed in the property view which can be
selected from the View menu or by pressing F4. Write the following
code and do not use the class builder wizard as it adds too much
code, more than we need for this simple class. The code is shown
below. You can add a form to test the code, but VB6 automatically
changes the instancing property and project properties when you do,
you must change them back after testing when you remove the form.
Compile the class by selecting from the menu File/Make dll.
Code for the COM Complex Class in VB6
Private mvre
Private mvim
Public Property Get Re()
Re = mvre
End Property
Public Property Get Im()
Im = mvim
End Property
Public Sub Create(real, imag)
mvre = real
mvim = imag
End Sub
This part of the class defines private internal values to hold
the real and imaginary parts of the complex number. The Create
subroutine is used to set these values. The values are made
available to the calling program using the property functions Get
Re and Get Im. These return the internal values of the real and
imaginary parts.
Public Function magnitude()
magnitude = Sqr(mvre * mvre + mvim * mvim)
End Function
Public Function FormatComplex(Optional n = 4) As String
FormatComplex = FormatNumber(mvre, n) & " + i" _
& "(" & FormatNumber(mvim, n) & ")"
End Function
The magnitude function in the class returns the magnitude of the
number, using the internal values. The format function returns a
string formatted to show the number in Cartesian format as 1.00 +
I(2.00). This uses an optional argument n with default value of 4.
If the calling program uses the syntax without the n, the program
will use 4 in its place. Functions must have the function name used
on the left side of an assignment statement to return the value to
the calling program.
Public Sub add(z As Complex)
mvre = mvre + z.Re
mvim = mvim + z.Im
End Sub
Public Sub mult(z As Complex)
Dim mvre1
mvre1 = mvre * z.Re - mvim * z.Im
mvim = mvre * z.Im + mvim * z.Re
mvre = mvre1
End Sub
Public Sub recip()
Dim a
a = mvre * mvre + mvim * mvim
mvre = mvre / a
mvim = mvim / a
End Sub
The operations of add, multiply and inverse are written as
subroutines to act on the internal values of the complex number.
The calling program will use these with syntax of z.add u where z
and u have been created as complex numbers. The real and imaginary
parts of the argument are used with the internal real and imaginary
parts to calculate the results that update the internal values.
Nothing is returned, only the internal values are modified. The Dim
statements indicate to the compiler that a new variable is in use.
If we use in VB6, the option explicit feature, this prevents an
error, otherwise the Dim is not necessary.
Map a Circle by Squaring
In our final project we will take points on a circle, square
them and visualize the resulting shape. We expect another circle,
but as we will see this is not so if the circle crosses the
imaginary axis. The COM will be modified to add a class for Circle.
The Circle class will be a complex number on the circle. The circle
will be created with a center at coordinates (a, b) with radius r.
We will write a method to return the center of the circle and one
to move the point to t radians on the circle. The Circle class will
use the complex number class and so is a dependent class. The
method of writing a dependent class is shown in the Circle class
code:
Class circle:
Private mvrecenter
Private mvimcenter
Private mvradius
Private mvz As Complex
Private Sub Class_Initialize()
Set mvz = New Complex
End Sub
Sub CCreate(a, b, r)
mvrecenter = A
mvimcenter = b
mvradius = r
mvz.Create real:=mvrecenter, imag:=mvimcenter
End Sub
The private variables maintain the internal state of the circle.
The complex class is connected to this class in the
Class_Initialize event which is called automatically when the
circle object is instantiated with a set statement. The Ccreate
method sets the center and radius of the internal values as a
complex object.
Function Eval(t) As Complex
Dim x
Dim y
Dim u As Complex
Set u = New Complex
x = mvradius * Cos(t)
y = mvradius * Sin(t)
u.Create real:=x, imag:=y
u.add mvz
Set Eval = u
End Function
The function Eval(t) returns a complex object with a value on
the circle at radian measure t.
Function CCenter()
Dim cc As Complex
Set cc = New Complex
cc.Create real:=mvrecenter, imag:=mvimcenter
Set CCenter = cc
End Function
The function Ccenter() returns the center of the circle and may
be used for documentation purposes.
A chart of the output is in Fig. 4.
1
2
3
4
5
6
7
8
9
10
11
A
B
C
D
E
F
G
H
I
J
K
=Witch2($B3,C$2) in each cell
Imag
0.000
0.000
0.500
1.000
1.500
2.000
2.500
3.000
3.500
4.000
0.000
1.000
1.333
0.800
0.333
0.190
0.125
0.089
0.067
0.500
0.800
0.894
0.970
0.555
0.294
0.179
0.120
0.087
0.065
1.000
0.500
0.496
0.447
0.332
0.224
0.152
0.108
0.081
0.062
Real
1.500
0.308
0.298
0.267
0.217
0.165
0.124
0.094
0.072
0.057
2.000
0.200
0.194
0.177
0.152
0.124
0.099
0.079
0.063
0.052
2.500
0.138
0.135
0.125
0.111
0.095
0.080
0.066
0.055
0.046
3.000
0.100
0.098
0.092
0.084
0.075
0.065
0.055
0.047
0.040
3.500
0.075
0.074
0.071
0.066
0.060
0.053
0.047
0.041
0.036
4.000
0.059
0.058
0.056
0.053
0.049
0.044
0.040
0.035
0.031
Fig. 4.
References
Lomax, Paul, VB & VBA In A Nutshell, OReilly, 1998
Walkenbach, John, Excel 2000 Programming, IDG, 1999
www.j-walk.com
EMBED Excel.Chart.8 \s
Array Enter the function into E2:F2 using Ctrl + Shift +
Enter.
{=Parallel(A2,B2,C2,D2)}
EMBED Excel.Sheet.8
_1002787200.xls
Chart1
0.4514851485
0.4615384615
0.2794636556
0.0827586207
-0.0827586207
-0.2117647059
-0.3084715715
-0.3787661406
-0.4282092948
-0.4615384615
-0.4826063348
-0.4944616748
-0.4994735372
-0.4994594595
-0.4958011421
-0.4895419188
-0.4814655588
-0.4721583065
-0.4620568746
-0.4514851485
-0.4406820495
-0.4298225915
-0.4190337562
-0.4084064477
-0.3980044954
-0.3878714372
-0.3780356352
-0.3685141374
-0.3593155965
-0.3504424779
-0.34189273
-0.3336610493
-0.3257398369
-0.3181199209
-0.3107911018
-0.3037425612
-0.2969631674
-0.2904417027
-0.2841670291
-0.2781282089
-0.27231459
-0.2667158643
-0.2613221055
-0.2561237924
-0.2511118198
-0.2462775015
-0.2416125666
-0.2371091508
-0.2327597859
-0.2285573851
-0.2244952285
-0.220566947
-0.2167665056
-0.2130881872
-0.2095265764
-0.2060765433
-0.2027332282
-0.1994920266
-0.1963485749
-0.1932987371
-0.1903385916
-0.1874644187
-0.1846726896
-0.1819600549
-0.1793233345
-0.1767595078
-0.1742657048
-0.171839197
-0.1694773896
-0.167177814
-0.1649381206
-0.1627560719
-0.1606295365
-0.1585564829
-0.1565349744
-0.1545631635
-0.1526392871
-0.1507616621
-0.1489286809
-0.1471388075
-0.1453905736
-0.1436825752
-0.1420134689
-0.1403819691
-0.138786845
-0.1372269174
-0.1357010567
-0.1342081798
-0.1327472483
-0.1313172658
-0.1299172763
-0.128546362
-0.1272036413
-0.1258882673
-0.1245994262
-0.1233363355
-0.1220982427
-0.1208844237
-0.119694182
-0.1185268467
BandPass with parameter Frequency
Parallel
Test of Complex Class
ZUParallelmag Pangle Pangle Deg
1.00004.000012.000013.00001.12883.21623.40851.233270.6597
4.00005.000013.000017.00003.05953.86424.92870.901151.6291
5.00006.000014.000021.00003.70554.68175.97060.901351.6386
6.00007.000015.000025.00004.35155.51197.02260.902551.7098
7.00008.000016.000029.00004.99846.35048.08150.904051.7935
8.00009.000017.000033.00005.64637.19429.14540.905451.8739
9.000010.000018.000037.00006.29518.041910.21270.906651.9465
10.000011.000019.000041.00006.94478.892211.28280.907852.0107
Parallel vs f
Parallel Impedances as a function of frequency
fZUParallelmag P
0.501.001.002.00-4.001.330.671.49
1.001.002.002.00-2.002.000.672.11
1.501.003.002.00-1.332.190.342.21
2.001.004.002.00-1.002.170.172.17
2.501.005.002.00-0.802.130.092.13
3.001.006.002.00-0.672.090.052.10
3.501.007.002.00-0.572.070.032.07
4.001.008.002.00-0.502.060.022.06
Z = 1 + i*f U = 2 - i*2/fParallel = 1/( 1/Z + 1/U )
BandPass
BandPass Filter Plot
Frequencyxyq=5
0.100.290.45
0.200.690.46
0.300.910.28
0.400.990.08
0.500.99-0.08
0.600.95-0.21
0.700.89-0.31
0.800.83-0.38
0.900.76-0.43
1.000.69-0.46
1.100.63-0.48
1.200.57-0.49
1.300.52-0.50
1.400.48-0.50
1.500.44-0.50
1.600.40-0.49
1.700.37-0.48
1.800.34-0.47
1.900.31-0.46
2.000.29-0.45
2.100.26-0.44
2.200.24-0.43
2.300.23-0.42
2.400.21-0.41
2.500.20-0.40
2.600.18-0.39
2.700.17-0.38
2.800.16-0.37
2.900.15-0.36
3.000.14-0.35
3.100.14-0.34
3.200.13-0.33
3.300.12-0.33
3.400.11-0.32
3.500.11-0.31
3.600.10-0.30
3.700.10-0.30
3.800.09-0.29
3.900.09-0.28
4.000.08-0.28
4.100.08-0.27
4.200.08-0.27
4.300.07-0.26
4.400.07-0.26
4.500.07-0.25
4.600.06-0.25
4.700.06-0.24
4.800.06-0.24
4.900.06-0.23
5.000.06-0.23
5.100.05-0.22
5.200.05-0.22
5.300.05-0.22
5.400.05-0.21
5.500.05-0.21
5.600.04-0.21
5.700.04-0.20
5.800.04-0.20
5.900.04-0.20
6.000.04-0.19
6.100.04-0.19
6.200.04-0.19
6.300.04-0.18
6.400.03-0.18
6.500.03-0.18
6.600.03-0.18
6.700.03-0.17
6.800.03-0.17
6.900.03-0.17
7.000.03-0.17
7.100.03-0.16
7.200.03-0.16
7.300.03-0.16
7.400.03-0.16
7.500.03-0.16
7.600.02-0.15
7.700.02-0.15
7.800.02-0.15
7.900.02-0.15
8.000.02-0.15
8.100.02-0.15
8.200.02-0.14
8.300.02-0.14
8.400.02-0.14
8.500.02-0.14
8.600.02-0.14
8.700.02-0.14
8.800.02-0.13
8.900.02-0.13
9.000.02-0.13
9.100.02-0.13
9.200.02-0.13
9.300.02-0.13
9.400.02-0.13
9.500.02-0.12
9.600.02-0.12
9.700.02-0.12
9.800.01-0.12
9.900.01-0.12
10.000.01-0.12
BandPass
&A
Page &P
BandPass with parameter Frequency
Estimate m
Derivative estimates at xo for displacements h
xo:1
hm
0.150-0.49993673
0.100-0.49998750
0.050-0.49999922
0.025-0.49999995
0.010-0.50000000
hxfyTanError
0.1500.8500.58060.57500.0056
0.1000.9000.55250.55000.0025
0.0500.9500.52560.52500.0006
0.0250.9750.51270.51250.0002
0.0100.9900.50500.50500.0000
0.0001.0000.50000.50000.0000
0.0101.0100.49500.49500.0000
0.0251.0250.48770.48750.0002
0.0501.0500.47560.47500.0006
0.1001.1000.45250.45000.0025
0.1501.1500.43060.42500.0056
BP = (1+q)/q * LP * HPLP = 1/( 1 + i*f )HP = i*q*f/( 1 +i*q*f
)
f(x) = 1/( 1 + x^2 )estimate m = Im( f(xo + i*h)/h )f'(1) =
-0.5yTan = 1 - x/2
Estimate m
00
00
00
00
00
00
00
00
00
00
00
3D Witch
3D Plot of Witch = 1/( 1 + z^2 )
Imag
0.0000.0000.5001.0001.5002.0002.5003.0003.5004.000
0.0001.0001.3330.8000.3330.1900.1250.0890.067
0.5000.8000.8940.9700.5550.2940.1790.1200.0870.065
1.0000.5000.4960.4470.3320.2240.1520.1080.0810.062
Real1.5000.3080.2980.2670.2170.1650.1240.0940.0720.057
2.0000.2000.1940.1770.1520.1240.0990.0790.0630.052
2.5000.1380.1350.1250.1110.0950.0800.0660.0550.046
3.0000.1000.0980.0920.0840.0750.0650.0550.0470.040
3.5000.0750.0740.0710.0660.0600.0530.0470.0410.036
4.0000.0590.0580.0560.0530.0490.0440.0400.0350.031
3D Witch
0.8944271910.49613893840.2981423970.19402850.13453455880.09802861630.07427813530.0580686705
0.97014250010.44721359550.26666666670.17677669530.12493900950.09245003270.07087699120.0559016994
0.55470019620.33218191940.21693045780.15151080380.11094003920.08419653360.06575959490.0525906223
0.29408584880.22360679770.16537964610.12403473460.09510340690.07453559920.059595380.048507125
0.17888543820.15238786360.12379689210.09922778770.07974522230.06467616670.05305581090.0440412199
0.12033136750.10846522890.09363291780.07905694150.06621754260.05547001960.04667282530.0395284708
0.08662961640.08056594950.07231015260.06342817210.05494422560.04734805490.04078236950.035211213
0.06543324260.06201736730.05711372490.05150262030.04580786670.04042260420.03554327110.0312347524
0.5
1
1.5
2
2.5
3
3.5
4
Witch
The Witch Function
ZW3D Plot of Witch
xyuv0.511.522.533.54
0.50.50.8-0.40.50.8944271910.97014250010.55470019620.29408584880.17888543820.12033136750.08662961640.0654332426
110.2-0.410.49613893840.44721359550.33218191940.22360679770.15238786360.10846522890.08056594950.0620173673
1.51.50.0470588235-0.21176470591.50.2981423970.26666666670.21693045780.16537964610.12379689210.09363291780.07231015260.0571137249
220.0153846154-0.123076923120.19402850.17677669530.15151080380.12403473460.09922778770.07905694150.06342817210.0515026203
2.52.50.0063593005-0.0794912562.50.13453455880.12493900950.11094003920.09510340690.07974522230.06621754260.05494422560.0458078667
330.0030769231-0.055384615430.09802861630.09245003270.08419653360.07453559920.06467616670.05547001960.04734805490.0404226042
3.53.50.0016632017-0.04074844073.50.07427813530.07087699120.06575959490.059595380.05305581090.04667282530.04078236950.0355432711
440.0009756098-0.031219512240.05806867050.05590169940.05259062230.0485071250.04404121990.03952847080.0352112130.0312347524
Derivative estimates at xo for various displacements h
xo:1
hm
0.1-0.499988
0.01-0.500000
0.001-0.500000
0.0001-0.500000
Fix imaginary part and vary the real part in Witch
z.imo:0.5Witch
z.Reuv
01.3333333333-0
0.21.1895798825-0.3011594639
0.40.9209594171-0.4048173262
0.60.6971923874-0.37686075
0.80.5404144473-0.3110298977
10.4307692308-0.2461538462
1.20.3511810266-0.1924279598
1.40.2912694404-0.1504712976
1.60.2448931275-0.1183773426
1.80.2082452597-0.0939452299
20.1788235294-0.0752941176
2.20.1548987062-0.0609619237
2.40.1352302966-0.0498544872
2.60.1189041816-0.0411652293
2.80.1052333694-0.0343019132
30.0936936937-0.0288288288
Witch
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
Circle Class
Circle centered at z=a+ib with radius r
Pi:3.1415926536a:1b:1r:1
ntthetauv
1.000000.100000.314162.093105.10793
2.000000.200000.628320.751485.74466
3.000000.300000.94248-0.751485.74466
4.000000.400001.25664-2.093105.10793
5.000000.500001.57080-3.000004.00000
6.000000.600001.88496-3.329162.69629
7.000000.700002.19911-3.102621.49141
8.000000.800002.51327-2.484590.60648
9.000000.900002.82743-1.711130.12814
10.000001.000003.14159-1.000000.00000
11.000001.100003.45575-0.475060.06764
12.000001.200003.76991-0.133450.15745
13.000001.300004.084070.133450.15745
14.000001.400004.398230.475060.06764
15.000001.500004.712391.000000.00000
16.000001.600005.026551.711130.12814
17.000001.700005.340712.484590.60648
18.000001.800005.654873.102621.49141
19.000001.900005.969033.329162.69629
20.000002.000006.283193.000004.00000
0.5
1
1.5
2
2.5
3
3.5
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Circle Class
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Circle Class (2)
Circle centered at z=a+ib with radius r under mapping w=z^2
Pi:3.1415926536a:1b:1r:1
ntthetauv
1.000.100.312.095.11
2.000.200.630.755.74
3.000.300.94-0.755.74
4.000.401.26-2.095.11
5.000.501.57-3.004.00
6.000.601.88-3.332.70
7.000.702.20-3.101.49
8.000.802.51-2.480.61
9.000.902.83-1.710.13
10.001.003.14-1.000.00
11.001.103.46-0.480.07
12.001.203.77-0.130.16
13.001.304.080.130.16
14.001.404.400.480.07
15.001.504.711.000.00
16.001.605.031.710.13
17.001.705.342.480.61
18.001.805.653.101.49
19.001.905.973.332.70
20.002.006.283.004.00
Circle Class (2)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Witch Circle on I (2)
Witch Plot on a Circle
a:1b:0r:1
tuvmag Pangle Pangle Deg
00.2-00.20000.00000.000015.5936
10.1990707248-0.0555577250.2067-0.2722-15.59360.0000
20.1963618866-0.11718205080.2287-0.5380-30.8273-15.2337
30.1925478056-0.19381341710.2732-0.7887-45.1877-29.5940
40.1916564754-0.30309019690.3586-1.0069-57.6931-42.0995
50.2175497984-0.48680519380.5332-1.1505-65.9205-50.3268
60.428649342-0.81733426410.9229-1.0878-62.3253-46.7317
71.205297595-0.73158411111.4099-0.5455-31.2567-15.6630
81.172371077-0.09153975181.1759-0.0779-4.464611.1290
91.0047493827-0.00032906921.0047-0.0003-0.018815.5749
101.07605917970.02302561811.07630.02141.225816.8195
111.30791751050.39240916111.36550.291516.700632.2942
120.6930822290.93458977851.16350.932753.439769.0334
130.25879321910.60048786560.65391.163966.685382.2790
140.19569929430.36440387090.41361.078061.762477.3561
150.19131839920.23241661750.30100.882150.539866.1335
160.19486577360.14558419170.24320.641636.763452.3570
170.19817906380.07928284520.21340.380621.804237.3978
180.19984838570.0220916380.20110.11016.308021.9016
190.1996683422-0.03279026380.2023-0.1628-9.32616.2676
200.1976516133-0.09113441850.2177-0.4320-24.7538-9.1601
Witch Circle on I (2)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Witch Circle on I
Witch Plot on a Circle
a:0b:1r:1
tuv
00.2-0.4mag Pangle Pangle Deg
10.0167440705-0.39681996950.3972-1.5286-87.5838-0.0000
2-0.1406555859-0.33633994360.3646-1.9669-112.6944-25.1106
3-0.2576106597-0.22899731310.3447-2.4149-138.3652-50.7814
4-0.3224584422-0.09042543180.3349-2.8682-164.3352-76.7514
5-0.32852478680.06035196340.33402.9599169.5905257.1743
6-0.27517914140.20299052690.34192.5060143.5850231.1688
7-0.16793069040.31802640120.35962.0566117.8359205.4197
8-0.01756827680.38898842570.38941.615992.5859180.1697
90.16169052350.40425644090.43541.190368.2001155.7839
100.35518815730.35879883990.50490.790545.2897132.8735
110.55218526920.25706357490.60910.435724.9638112.5476
120.74955726960.12216449310.75940.16169.256896.8406
130.93266626330.01756314370.93280.01881.078888.6626
140.9954849578-0.00030355130.9955-0.0003-0.017587.5663
150.8658103309-0.04928560770.8672-0.0569-3.258084.3258
160.6715658273-0.17653829580.6944-0.2571-14.728472.8554
170.4743164147-0.30311335910.5629-0.5686-32.580755.0031
180.2778829746-0.3839950610.4740-0.9444-54.108133.4757
190.0883959644-0.40537070980.4149-1.3561-77.69859.8853
20-0.0812566035-0.36709323370.3760-1.7886-102.4813-14.8975
Witch Circle on I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
_1002790369.xls
Parallel
Test of Complex Class
ZUParallelmag Pangle Pangle Deg
1.00004.000012.000013.00001.12883.21623.40851.233270.6597
4.00005.000013.000017.00003.05953.86424.92870.901151.6291
5.00006.000014.000021.00003.70554.68175.97060.901351.6386
6.00007.000015.000025.00004.35155.51197.02260.902551.7098
7.00008.000016.000029.00004.99846.35048.08150.904051.7935
8.00009.000017.000033.00005.64637.19429.14540.905451.8739
9.000010.000018.000037.00006.29518.041910.21270.906651.9465
10.000011.000019.000041.00006.94478.892211.28280.907852.0107
Parallel vs f
Parallel Impedances as a function of frequency
fZUParallelmag P
0.501.001.002.00-4.001.330.671.49
1.001.002.002.00-2.002.000.672.11
1.501.003.002.00-1.332.190.342.21
2.001.004.002.00-1.002.170.172.17
2.501.005.002.00-0.802.130.092.13
3.001.006.002.00-0.672.090.052.10
3.501.007.002.00-0.572.070.032.07
4.001.008.002.00-0.502.060.022.06
Z = 1 + i*f U = 2 - i*2/fParallel = 1/( 1/Z + 1/U )
BandPass
BandPass Filter Plot
Frequencyxyq=5
0.100.290.45
0.200.690.46
0.300.910.28
0.400.990.08
0.500.99-0.08
0.600.95-0.21
0.700.89-0.31
0.800.83-0.38
0.900.76-0.43
1.000.69-0.46
1.100.63-0.48
1.200.57-0.49
1.300.52-0.50
1.400.48-0.50
1.500.44-0.50
1.600.40-0.49
1.700.37-0.48
1.800.34-0.47
1.900.31-0.46
2.000.29-0.45
2.100.26-0.44
2.200.24-0.43
2.300.23-0.42
2.400.21-0.41
2.500.20-0.40
2.600.18-0.39
2.700.17-0.38
2.800.16-0.37
2.900.15-0.36
3.000.14-0.35
3.100.14-0.34
3.200.13-0.33
3.300.12-0.33
3.400.11-0.32
3.500.11-0.31
3.600.10-0.30
3.700.10-0.30
3.800.09-0.29
3.900.09-0.28
4.000.08-0.28
4.100.08-0.27
4.200.08-0.27
4.300.07-0.26
4.400.07-0.26
4.500.07-0.25
4.600.06-0.25
4.700.06-0.24
4.800.06-0.24
4.900.06-0.23
5.000.06-0.23
5.100.05-0.22
5.200.05-0.22
5.300.05-0.22
5.400.05-0.21
5.500.05-0.21
5.600.04-0.21
5.700.04-0.20
5.800.04-0.20
5.900.04-0.20
6.000.04-0.19
6.100.04-0.19
6.200.04-0.19
6.300.04-0.18
6.400.03-0.18
6.500.03-0.18
6.600.03-0.18
6.700.03-0.17
6.800.03-0.17
6.900.03-0.17
7.000.03-0.17
7.100.03-0.16
7.200.03-0.16
7.300.03-0.16
7.400.03-0.16
7.500.03-0.16
7.600.02-0.15
7.700.02-0.15
7.800.02-0.15
7.900.02-0.15
8.000.02-0.15
8.100.02-0.15
8.200.02-0.14
8.300.02-0.14
8.400.02-0.14
8.500.02-0.14
8.600.02-0.14
8.700.02-0.14
8.800.02-0.13
8.900.02-0.13
9.000.02-0.13
9.100.02-0.13
9.200.02-0.13
9.300.02-0.13
9.400.02-0.13
9.500.02-0.12
9.600.02-0.12
9.700.02-0.12
9.800.01-0.12
9.900.01-0.12
10.000.01-0.12
BandPass
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
&A
Page &P
BandPass with parameter Frequency
Sheet1
1
hm
0.150-0.49994
0.100-0.49999
0.050-0.50000
0.025-0.50000
0.010-0.50000
BP = (1+q)/q * LP * HPLP = 1/( 1 + i*f )HP = i*q*f/( 1 +i*q*f
)
Estimate m
Derivative estimates at xo for displacements h
xo:1
hm
0.150-0.49993673
0.100-0.49998750
0.050-0.49999922
0.025-0.49999995
0.010-0.50000000
hxfyTanError
0.1500.8500.58060.57500.0056
0.1000.9000.55250.55000.0025
0.0500.9500.52560.52500.0006
0.0250.9750.51270.51250.0002
0.0100.9900.50500.50500.0000
0.0001.0000.50000.50000.0000
0.0101.0100.49500.49500.0000
0.0251.0250.48770.48750.0002
0.0501.0500.47560.47500.0006
0.1001.1000.45250.45000.0025
0.1501.1500.43060.42500.0056
f(x) = 1/( 1 + x^2 )estimate m = Im( f(xo + i*h)/h )f'(1) =
-0.5yTan = 1 - x/2
3D Witch
3D Plot of Witch = 1/( 1 + z^2 )
Imag
0.0000.0000.5001.0001.5002.0002.5003.0003.5004.000
0.0001.0001.3330.8000.3330.1900.1250.0890.067
0.5000.8000.8940.9700.5550.2940.1790.1200.0870.065
1.0000.5000.4960.4470.3320.2240.1520.1080.0810.062
Real1.5000.3080.2980.2670.2170.1650.1240.0940.0720.057
2.0000.2000.1940.1770.1520.1240.0990.0790.0630.052
2.5000.1380.1350.1250.1110.0950.0800.0660.0550.046
3.0000.1000.0980.0920.0840.0750.0650.0550.0470.040
3.5000.0750.0740.0710.0660.0600.0530.0470.0410.036
4.0000.0590.0580.0560.0530.0490.0440.0400.0350.031
3D Witch
0.8944271910.49613893840.2981423970.19402850.13453455880.09802861630.07427813530.0580686705
0.97014250010.44721359550.26666666670.17677669530.12493900950.09245003270.07087699120.0559016994
0.55470019620.33218191940.21693045780.15151080380.11094003920.08419653360.06575959490.0525906223
0.29408584880.22360679770.16537964610.12403473460.09510340690.07453559920.059595380.048507125
0.17888543820.15238786360.12379689210.09922778770.07974522230.06467616670.05305581090.0440412199
0.12033136750.10846522890.09363291780.07905694150.06621754260.05547001960.04667282530.0395284708
0.08662961640.08056594950.07231015260.06342817210.05494422560.04734805490.04078236950.035211213
0.06543324260.06201736730.05711372490.05150262030.04580786670.04042260420.03554327110.0312347524
0.5
1
1.5
2
2.5
3
3.5
4
Witch
The Witch Function
ZW3D Plot of Witch
xyuv0.511.522.533.54
0.50.50.8-0.40.50.8944271910.97014250010.55470019620.29408584880.17888543820.12033136750.08662961640.0654332426
110.2-0.410.49613893840.44721359550.33218191940.22360679770.15238786360.10846522890.08056594950.0620173673
1.51.50.0470588235-0.21176470591.50.2981423970.26666666670.21693045780.16537964610.12379689210.09363291780.07231015260.0571137249
220.0153846154-0.123076923120.19402850.17677669530.15151080380.12403473460.09922778770.07905694150.06342817210.0515026203
2.52.50.0063593005-0.0794912562.50.13453455880.12493900950.11094003920.09510340690.07974522230.06621754260.05494422560.0458078667
330.0030769231-0.055384615430.09802861630.09245003270.08419653360.07453559920.06467616670.05547001960.04734805490.0404226042
3.53.50.0016632017-0.04074844073.50.07427813530.07087699120.06575959490.059595380.05305581090.04667282530.04078236950.0355432711
440.0009756098-0.031219512240.05806867050.05590169940.05259062230.0485071250.04404121990.03952847080.0352112130.0312347524
Derivative estimates at xo for various displacements h
xo:1
hm
0.1-0.499988
0.01-0.500000
0.001-0.500000
0.0001-0.500000
Fix imaginary part and vary the real part in Witch
z.imo:0.5Witch
z.Reuv
01.3333333333-0
0.21.1895798825-0.3011594639
0.40.9209594171-0.4048173262
0.60.6971923874-0.37686075
0.80.5404144473-0.3110298977
10.4307692308-0.2461538462
1.20.3511810266-0.1924279598
1.40.2912694404-0.1504712976
1.60.2448931275-0.1183773426
1.80.2082452597-0.0939452299
20.1788235294-0.0752941176
2.20.1548987062-0.0609619237
2.40.1352302966-0.0498544872
2.60.1189041816-0.0411652293
2.80.1052333694-0.0343019132
30.0936936937-0.0288288288
Witch
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
Circle Class
Circle centered at z=a+ib with radius r
Pi:3.1415926536a:1b:1r:1
ntthetauv
1.000000.100000.314162.093105.10793
2.000000.200000.628320.751485.74466
3.000000.300000.94248-0.751485.74466
4.000000.400001.25664-2.093105.10793
5.000000.500001.57080-3.000004.00000
6.000000.600001.88496-3.329162.69629
7.000000.700002.19911-3.102621.49141
8.000000.800002.51327-2.484590.60648
9.000000.900002.82743-1.711130.12814
10.000001.000003.14159-1.000000.00000
11.000001.100003.45575-0.475060.06764
12.000001.200003.76991-0.133450.15745
13.000001.300004.084070.133450.15745
14.000001.400004.398230.475060.06764
15.000001.500004.712391.000000.00000
16.000001.600005.026551.711130.12814
17.000001.700005.340712.484590.60648
18.000001.800005.654873.102621.49141
19.000001.900005.969033.329162.69629
20.000002.000006.283193.000004.00000
0.5
1
1.5
2
2.5
3
3.5
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Circle Class
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Circle Class (2)
Circle centered at z=a+ib with radius r under mapping w=z^2
Pi:3.1415926536a:1b:1r:1
ntthetauv
1.000.100.312.095.11
2.000.200.630.755.74
3.000.300.94-0.755.74
4.000.401.26-2.095.11
5.000.501.57-3.004.00
6.000.601.88-3.332.70
7.000.702.20-3.101.49
8.000.802.51-2.480.61
9.000.902.83-1.710.13
10.001.003.14-1.000.00
11.001.103.46-0.480.07
12.001.203.77-0.130.16
13.001.304.080.130.16
14.001.404.400.480.07
15.001.504.711.000.00
16.001.605.031.710.13
17.001.705.342.480.61
18.001.805.653.101.49
19.001.905.973.332.70
20.002.006.283.004.00
Circle Class (2)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Witch Circle on I (2)
Witch Plot on a Circle
a:1b:0r:1
tuvmag Pangle Pangle Deg
00.2-00.20000.00000.000015.5936
10.1990707248-0.0555577250.2067-0.2722-15.59360.0000
20.1963618866-0.11718205080.2287-0.5380-30.8273-15.2337
30.1925478056-0.19381341710.2732-0.7887-45.1877-29.5940
40.1916564754-0.30309019690.3586-1.0069-57.6931-42.0995
50.2175497984-0.48680519380.5332-1.1505-65.9205-50.3268
60.428649342-0.81733426410.9229-1.0878-62.3253-46.7317
71.205297595-0.73158411111.4099-0.5455-31.2567-15.6630
81.172371077-0.09153975181.1759-0.0779-4.464611.1290
91.0047493827-0.00032906921.0047-0.0003-0.018815.5749
101.07605917970.02302561811.07630.02141.225816.8195
111.30791751050.39240916111.36550.291516.700632.2942
120.6930822290.93458977851.16350.932753.439769.0334
130.25879321910.60048786560.65391.163966.685382.2790
140.19569929430.36440387090.41361.078061.762477.3561
150.19131839920.23241661750.30100.882150.539866.1335
160.19486577360.14558419170.24320.641636.763452.3570
170.19817906380.07928284520.21340.380621.804237.3978
180.19984838570.0220916380.20110.11016.308021.9016
190.1996683422-0.03279026380.2023-0.1628-9.32616.2676
200.1976516133-0.09113441850.2177-0.4320-24.7538-9.1601
Witch Circle on I (2)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Witch Circle on I
Witch Plot on a Circle
a:0b:1r:1
tuv
00.2-0.4mag Pangle Pangle Deg
10.0167440705-0.39681996950.3972-1.5286-87.5838-0.0000
2-0.1406555859-0.33633994360.3646-1.9669-112.6944-25.1106
3-0.2576106597-0.22899731310.3447-2.4149-138.3652-50.7814
4-0.3224584422-0.09042543180.3349-2.8682-164.3352-76.7514
5-0.32852478680.06035196340.33402.9599169.5905257.1743
6-0.27517914140.20299052690.34192.5060143.5850231.1688
7-0.16793069040.31802640120.35962.0566117.8359205.4197
8-0.01756827680.38898842570.38941.615992.5859180.1697
90.16169052350.40425644090.43541.190368.2001155.7839
100.35518815730.35879883990.50490.790545.2897132.8735
110.55218526920.25706357490.60910.435724.9638112.5476
120.74955726960.12216449310.75940.16169.256896.8406
130.93266626330.01756314370.93280.01881.078888.6626
140.9954849578-0.00030355130.9955-0.0003-0.017587.5663
150.8658103309-0.04928560770.8672-0.0569-3.258084.3258
160.6715658273-0.17653829580.6944-0.2571-14.728472.8554
170.4743164147-0.30311335910.5629-0.5686-32.580755.0031
180.2778829746-0.3839950610.4740-0.9444-54.108133.4757
190.0883959644-0.40537070980.4149-1.3561-77.69859.8853
20-0.0812566035-0.36709323370.3760-1.7886-102.4813-14.8975
Witch Circle on I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
_1002955649.xls
Chart1
0.8944271910.49613893840.2981423970.19402850.13453455880.09802861630.07427813530.0580686705
0.97014250010.44721359550.26666666670.17677669530.12493900950.09245003270.07087699120.0559016994
0.55470019620.33218191940.21693045780.15151080380.11094003920.08419653360.06575959490.0525906223
0.29408584880.22360679770.16537964610.12403473460.09510340690.07453559920.059595380.048507125
0.17888543820.15238786360.12379689210.09922778770.07974522230.06467616670.05305581090.0440412199
0.12033136750.10846522890.09363291780.07905694150.06621754260.05547001960.04667282530.0395284708
0.08662961640.08056594950.07231015260.06342817210.05494422560.04734805490.04078236950.035211213
0.06543324260.06201736730.05711372490.05150262030.04580786670.04042260420.03554327110.0312347524
0.5
1
1.5
2
2.5
3
3.5
4
Parallel
Test of Complex Class
ZUParallelmag Pangle Pangle Deg
1.00004.000012.000013.00001.12883.21623.40851.233270.6597
4.00005.000013.000017.00003.05953.86424.92870.901151.6291
5.00006.000014.000021.00003.70554.68175.97060.901351.6386
6.00007.000015.000025.00004.35155.51197.02260.902551.7098
7.00008.000016.000029.00004.99846.35048.08150.904051.7935
8.00009.000017.000033.00005.64637.19429.14540.905451.8739
9.000010.000018.000037.00006.29518.041910.21270.906651.9465
10.000011.000019.000041.00006.94478.892211.28280.907852.0107
Parallel vs f
Parallel Impedances as a function of frequency
fZUParallelmag P
0.501.001.002.00-4.001.330.671.49
1.001.002.002.00-2.002.000.672.11
1.501.003.002.00-1.332.190.342.21
2.001.004.002.00-1.002.170.172.17
2.501.005.002.00-0.802.130.092.13
3.001.006.002.00-0.672.090.052.10
3.501.007.002.00-0.572.070.032.07
4.001.008.002.00-0.502.060.022.06
Z = 1 + i*f U = 2 - i*2/fParallel = 1/( 1/Z + 1/U )
BandPass
BandPass Filter Plot
Frequencyxyq=5
0.100.290.45
0.200.690.46
0.300.910.28
0.400.990.08
0.500.99-0.08
0.600.95-0.21
0.700.89-0.31
0.800.83-0.38
0.900.76-0.43
1.000.69-0.46
1.100.63-0.48
1.200.57-0.49
1.300.52-0.50
1.400.48-0.50
1.500.44-0.50
1.600.40-0.49
1.700.37-0.48
1.800.34-0.47
1.900.31-0.46
2.000.29-0.45
2.100.26-0.44
2.200.24-0.43
2.300.23-0.42
2.400.21-0.41
2.500.20-0.40
2.600.18-0.39
2.700.17-0.38
2.800.16-0.37
2.900.15-0.36
3.000.14-0.35
3.100.14-0.34
3.200.13-0.33
3.300.12-0.33
3.400.11-0.32
3.500.11-0.31
3.600.10-0.30
3.700.10-0.30
3.800.09-0.29
3.900.09-0.28
4.000.08-0.28
4.100.08-0.27
4.200.08-0.27
4.300.07-0.26
4.400.07-0.26
4.500.07-0.25
4.600.06-0.25
4.700.06-0.24
4.800.06-0.24
4.900.06-0.23
5.000.06-0.23
5.100.05-0.22
5.200.05-0.22
5.300.05-0.22
5.400.05-0.21
5.500.05-0.21
5.600.04-0.21
5.700.04-0.20
5.800.04-0.20
5.900.04-0.20
6.000.04-0.19
6.100.04-0.19
6.200.04-0.19
6.300.04-0.18
6.400.03-0.18
6.500.03-0.18
6.600.03-0.18
6.700.03-0.17
6.800.03-0.17
6.900.03-0.17
7.000.03-0.17
7.100.03-0.16
7.200.03-0.16
7.300.03-0.16
7.400.03-0.16
7.500.03-0.16
7.600.02-0.15
7.700.02-0.15
7.800.02-0.15
7.900.02-0.15
8.000.02-0.15
8.100.02-0.15
8.200.02-0.14
8.300.02-0.14
8.400.02-0.14
8.500.02-0.14
8.600.02-0.14
8.700.02-0.14
8.800.02-0.13
8.900.02-0.13
9.000.02-0.13
9.100.02-0.13
9.200.02-0.13
9.300.02-0.13
9.400.02-0.13
9.500.02-0.12
9.600.02-0.12
9.700.02-0.12
9.800.01-0.12
9.900.01-0.12
10.000.01-0.12
BandPass
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
&A
Page &P
BandPass with parameter Frequency
Estimate m
Derivative estimates at xo for displacements h
xo:1
hm
0.150-0.49993673
0.100-0.49998750
0.050-0.49999922
0.025-0.49999995
0.010-0.50000000
hxfyTanError
0.1500.8500.58060.57500.0056
0.1000.9000.55250.55000.0025
0.0500.9500.52560.52500.0006
0.0250.9750.51270.51250.0002
0.0100.9900.50500.50500.0000
0.0001.0000.50000.50000.0000
0.0101.0100.49500.49500.0000
0.0251.0250.48770.48750.0002
0.0501.0500.47560.47500.0006
0.1001.1000.45250.45000.0025
0.1501.1500.43060.42500.0056
BP = (1+q)/q * LP * HPLP = 1/( 1 + i*f )HP = i*q*f/( 1 +i*q*f
)
f(x) = 1/( 1 + x^2 )estimate m = Im( f(xo + i*h)/h )f'(1) =
-0.5yTan = 1 - x/2
Estimate m
00
00
00
00
00
00
00
00
00
00
00
3D Witch
3D Plot of Witch = 1/( 1 + z^2 )
Imag
0.0000.0000.5001.0001.5002.0002.5003.0003.5004.000
0.0001.0001.3330.8000.3330.1900.1250.0890.067
0.5000.8000.8940.9700.5550.2940.1790.1200.0870.065
1.0000.5000.4960.4470.3320.2240.1520.1080.0810.062
Real1.5000.3080.2980.2670.2170.1650.1240.0940.0720.057
2.0000.2000.1940.1770.1520.1240.0990.0790.0630.052
2.5000.1380.1350.1250.1110.0950.0800.0660.0550.046
3.0000.1000.0980.0920.0840.0750.0650.0550.0470.040
3.5000.0750.0740.0710.0660.0600.0530.0470.0410.036
4.0000.0590.0580.0560.0530.0490.0440.0400.0350.031
3D Witch
0.8944271910.49613893840.2981423970.19402850.13453455880.09802861630.07427813530.0580686705
0.97014250010.44721359550.26666666670.17677669530.12493900950.09245003270.07087699120.0559016994
0.55470019620.33218191940.21693045780.15151080380.11094003920.08419653360.06575959490.0525906223
0.29408584880.22360679770.16537964610.12403473460.09510340690.07453559920.059595380.048507125
0.17888543820.15238786360.12379689210.09922778770.07974522230.06467616670.05305581090.0440412199
0.12033136750.10846522890.09363291780.07905694150.06621754260.05547001960.04667282530.0395284708
0.08662961640.08056594950.07231015260.06342817210.05494422560.04734805490.04078236950.035211213
0.06543324260.06201736730.05711372490.05150262030.04580786670.04042260420.03554327110.0312347524
0.5
1
1.5
2
2.5
3
3.5
4
Witch
The Witch Function
ZW3D Plot of Witch
xyuv0.511.522.533.54
0.50.50.8-0.40.50.8944271910.97014250010.55470019620.29408584880.17888543820.12033136750.08662961640.0654332426
110.2-0.410.49613893840.44721359550.33218191940.22360679770.15238786360.10846522890.08056594950.0620173673
1.51.50.0470588235-0.21176470591.50.2981423970.26666666670.21693045780.16537964610.12379689210.09363291780.07231015260.0571137249
220.0153846154-0.123076923120.19402850.17677669530.15151080380.12403473460.09922778770.07905694150.06342817210.0515026203
2.52.50.0063593005-0.0794912562.50.13453455880.12493900950.11094003920.09510340690.07974522230.06621754260.05494422560.0458078667
330.0030769231-0.055384615430.09802861630.09245003270.08419653360.07453559920.06467616670.05547001960.04734805490.0404226042
3.53.50.0016632017-0.04074844073.50.07427813530.07087699120.06575959490.059595380.05305581090.04667282530.04078236950.0355432711
440.0009756098-0.031219512240.05806867050.05590169940.05259062230.0485071250.04404121990.03952847080.0352112130.0312347524
Derivative estimates at xo for various displacements h
xo:1
hm
0.1-0.499988
0.01-0.500000
0.001-0.500000
0.0001-0.500000
Fix imaginary part and vary the real part in Witch
z.imo:0.5Witch
z.Reuv
01.3333333333-0
0.21.1895798825-0.3011594639
0.40.9209594171-0.4048173262
0.60.6971923874-0.37686075
0.80.5404144473-0.3110298977
10.4307692308-0.2461538462
1.20.3511810266-0.1924279598
1.40.2912694404-0.1504712976
1.60.2448931275-0.1183773426
1.80.2082452597-0.0939452299
20.1788235294-0.0752941176
2.20.1548987062-0.0609619237
2.40.1352302966-0.0498544872
2.60.1189041816-0.0411652293
2.80.1052333694-0.0343019132
30.0936936937-0.0288288288
Witch
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
Circle Class
Circle centered at z=a+ib with radius r
Pi:3.1415926536a:1b:1r:1
ntthetauv
1.000000.100000.314162.093105.10793
2.000000.200000.628320.751485.74466
3.000000.300000.94248-0.751485.74466
4.000000.400001.25664-2.093105.10793
5.000000.500001.57080-3.000004.00000
6.000000.600001.88496-3.329162.69629
7.000000.700002.19911-3.102621.49141
8.000000.800002.51327-2.484590.60648
9.000000.900002.82743-1.711130.12814
10.000001.000003.14159-1.000000.00000
11.000001.100003.45575-0.475060.06764
12.000001.200003.76991-0.133450.15745
13.000001.300004.084070.133450.15745
14.000001.400004.398230.475060.06764
15.000001.500004.712391.000000.00000
16.000001.600005.026551.711130.12814
17.000001.700005.340712.484590.60648
18.000001.800005.654873.102621.49141
19.000001.900005.969033.329162.69629
20.000002.000006.283193.000004.00000
0.5
1
1.5
2
2.5
3
3.5
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Circle Class
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Circle Class (2)
Circle centered at z=a+ib with radius r under mapping w=z^2
Pi:3.1415926536a:1b:1r:1
ntthetauv
1.000.100.312.095.11
2.000.200.630.755.74
3.000.300.94-0.755.74
4.000.401.26-2.095.11
5.000.501.57-3.004.00
6.000.601.88-3.332.70
7.000.702.20-3.101.49
8.000.802.51-2.480.61
9.000.902.83-1.710.13
10.001.003.14-1.000.00
11.001.103.46-0.480.07
12.001.203.77-0.130.16
13.001.304.080.130.16
14.001.404.400.480.07
15.001.504.711.000.00
16.001.605.031.710.13
17.001.705.342.480.61
18.001.805.653.101.49
19.001.905.973.332.70
20.002.006.283.004.00
Circle Class (2)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Witch Circle on I (2)
Witch Plot on a Circle
a:1b:0r:1
tuvmag Pangle Pangle Deg
00.2-00.20000.00000.000015.5936
10.1990707248-0.0555577250.2067-0.2722-15.59360.0000
20.1963618866-0.11718205080.2287-0.5380-30.8273-15.2337
30.1925478056-0.19381341710.2732-0.7887-45.1877-29.5940
40.1916564754-0.30309019690.3586-1.0069-57.6931-42.0995
50.2175497984-0.48680519380.5332-1.1505-65.9205-50.3268
60.428649342-0.81733426410.9229-1.0878-62.3253-46.7317
71.205297595-0.73158411111.4099-0.5455-31.2567-15.6630
81.172371077-0.09153975181.1759-0.0779-4.464611.1290
91.0047493827-0.00032906921.0047-0.0003-0.018815.5749
101.07605917970.02302561811.07630.02141.225816.8195
111.30791751050.39240916111.36550.291516.700632.2942
120.6930822290.93458977851.16350.932753.439769.0334
130.25879321910.60048786560.65391.163966.685382.2790
140.19569929430.36440387090.41361.078061.762477.3561
150.19131839920.23241661750.30100.882150.539866.1335
160.19486577360.14558419170.24320.641636.763452.3570
170.19817906380.07928284520.21340.380621.804237.3978
180.19984838570.0220916380.20110.11016.308021.9016
190.1996683422-0.03279026380.2023-0.1628-9.32616.2676
200.1976516133-0.09113441850.2177-0.4320-24.7538-9.1601
Witch Circle on I (2)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Witch Circle on I
Witch Plot on a Circle
a:0b:1r:1
tuv
00.2-0.4mag Pangle Pangle Deg
10.0167440705-0.39681996950.3972-1.5286-87.5838-0.0000
2-0.1406555859-0.33633994360.3646-1.9669-112.6944-25.1106
3-0.2576106597-0.22899731310.3447-2.4149-138.3652-50.7814
4-0.3224584422-0.09042543180.3349-2.8682-164.3352-76.7514
5-0.32852478680.06035196340.33402.9599169.5905257.1743
6-0.27517914140.20299052690.34192.5060143.5850231.1688
7-0.16793069040.31802640120.35962.0566117.8359205.4197
8-0.01756827680.38898842570.38941.615992.5859180.1697
90.16169052350.40425644090.43541.190368.2001155.7839
100.35518815730.35879883990.50490.790545.2897132.8735
110.55218526920.25706357490.60910.435724.9638112.5476
120.74955726960.12216449310.75940.16169.256896.8406
130.93266626330.01756314370.93280.01881.078888.6626
140.9954849578-0.00030355130.9955-0.0003-0.017587.5663
150.8658103309-0.04928560770.8672-0.0569-3.258084.3258
160.6715658273-0.17653829580.6944-0.2571-14.728472.8554
170.4743164147-0.30311335910.5629-0.5686-32.580755.0031
180.2778829746-0.3839950610.4740-0.9444-54.108133.4757
190.0883959644-0.40537070980.4149-1.3561-77.69859.8853
20-0.0812566035-0.36709323370.3760-1.7886-102.4813-14.8975
Witch Circle on I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
_1002789964.xls
Chart2
0.58055152390.575
0.55248618780.55
0.52562417870.525
0.51265619990.5125
0.50502499870.505
0.50.5
0.49502499880.495
0.48765620240.4875
0.47562425680.475
0.45248868780.45
0.43057050590.425
Parallel
Test of Complex Class
ZUParallelmag Pangle Pangle Deg
1.00004.000012.000013.00001.12883.21623.40851.233270.6597
4.00005.000013.000017.00003.05953.86424.92870.901151.6291
5.00006.000014.000021.00003.70554.68175.97060.901351.6386
6.00007.000015.000025.00004.35155.51197.02260.902551.7098
7.00008.000016.000029.00004.99846.35048.08150.904051.7935
8.00009.000017.000033.00005.64637.19429.14540.905451.8739
9.000010.000018.000037.00006.29518.041910.21270.906651.9465
10.000011.000019.000041.00006.94478.892211.28280.907852.0107
Parallel vs f
Parallel Impedances as a function of frequency
fZUParallelmag P
0.501.001.002.00-4.001.330.671.49
1.001.002.002.00-2.002.000.672.11
1.501.003.002.00-1.332.190.342.21
2.001.004.002.00-1.002.170.172.17
2.501.005.002.00-0.802.130.092.13
3.001.006.002.00-0.672.090.052.10
3.501.007.002.00-0.572.070.032.07
4.001.008.002.00-0.502.060.022.06
Z = 1 + i*f U = 2 - i*2/fParallel = 1/( 1/Z + 1/U )
BandPass
BandPass Filter Plot
Frequencyxyq=5
0.100.290.45
0.200.690.46
0.300.910.28
0.400.990.08
0.500.99-0.08
0.600.95-0.21
0.700.89-0.31
0.800.83-0.38
0.900.76-0.43
1.000.69-0.46
1.100.63-0.48
1.200.57-0.49
1.300.52-0.50
1.400.48-0.50
1.500.44-0.50
1.600.40-0.49
1.700.37-0.48
1.800.34-0.47
1.900.31-0.46
2.000.29-0.45
2.100.26-0.44
2.200.24-0.43
2.300.23-0.42
2.400.21-0.41
2.500.20-0.40
2.600.18-0.39
2.700.17-0.38
2.800.16-0.37
2.900.15-0.36
3.000.14-0.35
3.100.14-0.34
3.200.13-0.33
3.300.12-0.33
3.400.11-0.32
3.500.11-0.31
3.600.10-0.30
3.700.10-0.30
3.800.09-0.29
3.900.09-0.28
4.000.08-0.28
4.100.08-0.27
4.200.08-0.27
4.300.07-0.26
4.400.07-0.26
4.500.07-0.25
4.600.06-0.25
4.700.06-0.24
4.800.06-0.24
4.900.06-0.23
5.000.06-0.23
5.100.05-0.22
5.200.05-0.22
5.300.05-0.22
5.400.05-0.21
5.500.05-0.21
5.600.04-0.21
5.700.04-0.20
5.800.04-0.20
5.900.04-0.20
6.000.04-0.19
6.100.04-0.19
6.200.04-0.19
6.300.04-0.18
6.400.03-0.18
6.500.03-0.18
6.600.03-0.18
6.700.03-0.17
6.800.03-0.17
6.900.03-0.17
7.000.03-0.17
7.100.03-0.16
7.200.03-0.16
7.300.03-0.16
7.400.03-0.16
7.500.03-0.16
7.600.02-0.15
7.700.02-0.15
7.800.02-0.15
7.900.02-0.15
8.000.02-0.15
8.100.02-0.15
8.200.02-0.14
8.300.02-0.14
8.400.02-0.14
8.500.02-0.14
8.600.02-0.14
8.700.02-0.14
8.800.02-0.13
8.900.02-0.13
9.000.02-0.13
9.100.02-0.13
9.200.02-0.13
9.300.02-0.13
9.400.02-0.13
9.500.02-0.12
9.600.02-0.12
9.700.02-0.12
9.800.01-0.12
9.900.01-0.12
10.000.01-0.12
BandPass
&A
Page &P
BandPass with parameter Frequency
Estimate m
Derivative estimates at xo for displacements h
xo:1
hm
0.150-0.49993673
0.100-0.49998750
0.050-0.49999922
0.025-0.49999995
0.010-0.50000000
hxfyTanError
0.1500.8500.58060.57500.0056
0.1000.9000.55250.55000.0025
0.0500.9500.52560.52500.0006
0.0250.9750.51270.51250.0002
0.0100.9900.50500.50500.0000
0.0001.0000.50000.50000.0000
0.0101.0100.49500.49500.0000
0.0251.0250.48770.48750.0002
0.0501.0500.47560.47500.0006
0.1001.1000.45250.45000.0025
0.1501.1500.43060.42500.0056
BP = (1+q)/q * LP * HPLP = 1/( 1 + i*f )HP = i*q*f/( 1 +i*q*f
)
f(x) = 1/( 1 + x^2 )estimate m = Im( f(xo + i*h)/h )f'(1) =
-0.5yTan = 1 - x/2
3D Witch
3D Plot of Witch = 1/( 1 + z^2 )
Imag
0.0000.0000.5001.0001.5002.0002.5003.0003.5004.000
0.0001.0001.3330.8000.3330.1900.1250.0890.067
0.5000.8000.8940.9700.5550.2940.1790.1200.0870.065
1.0000.5000.4960.4470.3320.2240.1520.1080.0810.062
Real1.5000.3080.2980.2670.2170.1650.1240.0940.0720.057
2.0000.2000.1940.1770.1520.1240.0990.0790.0630.052
2.5000.1380.1350.1250.1110.0950.0800.0660.0550.046
3.0000.1000.0980.0920.0840.0750.0650.0550.0470.040
3.5000.0750.0740.0710.0660.0600.0530.0470.0410.036
4.0000.0590.0580.0560.0530.0490.0440.0400.0350.031
3D Witch
0.8944271910.49613893840.2981423970.19402850.13453455880.09802861630.07427813530.0580686705
0.97014250010.44721359550.26666666670.17677669530.12493900950.09245003270.07087699120.0559016994
0.55470019620.33218191940.21693045780.15151080380.11094003920.08419653360.06575959490.0525906223
0.29408584880.22360679770.16537964610.12403473460.09510340690.07453559920.059595380.048507125
0.17888543820.15238786360.12379689210.09922778770.07974522230.06467616670.05305581090.0440412199
0.12033136750.10846522890.09363291780.07905694150.06621754260.05547001960.04667282530.0395284708
0.08662961640.08056594950.07231015260.06342817210.05494422560.04734805490.04078236950.035211213
0.06543324260.06201736730.05711372490.05150262030.04580786670.04042260420.03554327110.0312347524
0.5
1
1.5
2
2.5
3
3.5
4
Witch
The Witch Function
ZW3D Plot of Witch
xyuv0.511.522.533.54
0.50.50.8-0.40.50.8944271910.97014250010.55470019620.29408584880.17888543820.12033136750.08662961640.0654332426
110.2-0.410.49613893840.44721359550.33218191940.22360679770.15238786360.10846522890.08056594950.0620173673
1.51.50.0470588235-0.21176470591.50.2981423970.26666666670.21693045780.16537964610.12379689210.09363291780.07231015260.0571137249
220.0153846154-0.123076923120.19402850.17677669530.15151080380.12403473460.09922778770.07905694150.06342817210.0515026203
2.52.50.0063593005-0.0794912562.50.13453455880.12493900950.11094003920.09510340690.07974522230.06621754260.05494422560.0458078667
330.0030769231-0.055384615430.09802861630.09245003270.08419653360.07453559920.06467616670.05547001960.04734805490.0404226042
3.53.50.0016632017-0.04074844073.50.07427813530.07087699120.06575959490.059595380.05305581090.04667282530.04078236950.0355432711
440.0009756098-0.031219512240.05806867050.05590169940.05259062230.0485071250.04404121990.03952847080.0352112130.0312347524
Derivative estimates at xo for various displacements h
xo:1
hm
0.1-0.499988
0.01-0.500000
0.001-0.500000
0.0001-0.500000
Fix imaginary part and vary the real part in Witch
z.imo:0.5Witch
z.Reuv
01.3333333333-0
0.21.1895798825-0.3011594639
0.40.9209594171-0.4048173262
0.60.6971923874-0.37686075
0.80.5404144473-0.3110298977
10.4307692308-0.2461538462
1.20.3511810266-0.1924279598
1.40.2912694404-0.1504712976
1.60.2448931275-0.1183773426
1.80.2082452597-0.0939452299
20.1788235294-0.0752941176
2.20.1548987062-0.0609619237
2.40.1352302966-0.0498544872
2.60.1189041816-0.0411652293
2.80.1052333694-0.0343019132
30.0936936937-0.0288288288
Witch
00000000
00000000
00000000
00000000
00000000
00000000
00000000
00000000
Circle Class
Circle centered at z=a+ib with radius r
Pi:3.1415926536a:1b:1r:1
ntthetauv
1.000000.100000.314162.093105.10793
2.000000.200000.628320.751485.74466
3.000000.300000.94248-0.751485.74466
4.000000.400001.25664-2.093105.10793
5.000000.500001.57080-3.000004.00000
6.000000.600001.88496-3.329162.69629
7.000000.700002.19911-3.102621.49141
8.000000.800002.51327-2.484590.60648
9.000000.900002.82743-1.711130.12814
10.000001.000003.14159-1.000000.00000
11.000001.100003.45575-0.475060.06764
12.000001.200003.76991-0.133450.15745
13.000001.300004.084070.133450.15745
14.000001.400004.398230.475060.06764
15.000001.500004.712391.000000.00000
16.000001.600005.026551.711130.12814
17.000001.700005.340712.484590.60648
18.000001.800005.654873.102621.49141
19.000001.900005.969033.329162.69629
20.000002.000006.283193.000004.00000
0.5
1
1.5
2
2.5
3
3.5
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Circle Class
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Circle Class (2)
Circle centered at z=a+ib with radius r under mapping w=z^2
Pi:3.1415926536a:1b:1r:1
ntthetauv
1.000.100.312.095.11
2.000.200.630.755.74
3.000.300.94-0.755.74
4.000.401.26-2.095.11
5.000.501.57-3.004.00
6.000.601.88-3.332.70
7.000.702.20-3.101.49
8.000.802.51-2.480.61
9.000.902.83-1.710.13
10.001.003.14-1.000.00
11.001.103.46-0.480.07
12.001.203.77-0.130.16
13.001.304.080.130.16
14.001.404.400.480.07
15.001.504.711.000.00
16.001.605.031.710.13
17.001.705.342.480.61
18.001.805.653.101.49
19.001.905.973.332.70
20.002.006.283.004.00
Circle Class (2)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Witch Circle on I (2)
Witch Plot on a Circle
a:1b:0r:1
tuvmag Pangle Pangle Deg
00.2-00.20000.00000.000015.5936
10.1990707248-0.0555577250.2067-0.2722-15.59360.0000
20.1963618866-0.11718205080.2287-0.5380-30.8273-15.2337
30.1925478056-0.19381341710.2732-0.7887-45.1877-29.5940
40.1916564754-0.30309019690.3586-1.0069-57.6931-42.0995
50.2175497984-0.48680519380.5332-1.1505-65.9205-50.3268
60.428649342-0.81733426410.9229-1.0878-62.3253-46.7317
71.205297595-0.73158411111.4099-0.5455-31.2567-15.6630
81.172371077-0.09153975181.1759-0.0779-4.464611.1290
91.0047493827-0.00032906921.0047-0.0003-0.018815.5749
101.07605917970.02302561811.07630.02141.225816.8195
111.30791751050.39240916111.36550.291516.700632.2942
120.6930822290.93458977851.16350.932753.439769.0334
130.25879321910.60048786560.65391.163966.685382.2790
140.19569929430.36440387090.41361.078061.762477.3561
150.19131839920.23241661750.30100.882150.539866.1335
160.19486577360.14558419170.24320.641636.763452.3570
170.19817906380.07928284520.21340.380621.804237.3978
180.19984838570.0220916380.20110.11016.308021.9016
190.1996683422-0.03279026380.2023-0.1628-9.32616.2676
200.1976516133-0.09113441850.2177-0.4320-24.7538-9.1601
Witch Circle on I (2)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Witch Circle on I
Witch Plot on a Circle
a:0b:1r:1
tuv
00.2-0.4mag Pangle Pangle Deg
10.0167440705-0.39681996950.3972-1.5286-87.5838-0.0000
2-0.1406555859-0.33633994360.3646-1.9669-112.6944-25.1106
3-0.2576106597-0.22899731310.3447-2.4149-138.3652-50.7814
4-0.3224584422-0.09042543180.3349-2.8682-164.3352-76.7514
5-0.32852478680.06035196340.33402.9599169.5905257.1743
6-0.27517914140.20299052690.34192.5060143.5850231.1688
7-0.16793069040.31802640120.35962.0566117.8359205.4197
8-0.01756827680.38898842570.38941.615992.5859180.1697
90.16169052350.40425644090.43541.190368.2001155.7839
100.35518815730.35879883990.50490.790545.2897132.8735
110.55218526920.25706357490.60910.435724.9638112.5476
120.74955726960.12216449310.75940.16169.256896.8406
130.93266626330.01756314370.93280.01881.078888.6626
140.9954849578-0.00030355130.9955-0.0003-0.017587.5663
150.8658103309-0.04928560770.8672-0.0569-3.258084.3258
160.6715658273-0.17653829580.6944-0.2571-14.728472.8554
170.4743164147-0.30311335910.5629-0.5686-32.580755.0031
180.2778829746-0.3839950610.4740-0.9444-54.108133.4757
190.0883959644-0.40537070980.4149-1.3561-77.69859.8853
20-0.0812566035-0.36709323370.3760-1.7886-102.4813-14.8975
Witch Circle on I
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
_1002785124.xls
Chart1
0.4562682136
0.8956603972
1.4803106183
2.3568938893
3.946809812
7.9199116157
19.8980049251
9.9739044126
5.607375353
3.946809812
3.0837103727
2.5530208318
2.1918476435
1.9288983748
1.728090448
1.5691977921
1.4399900448
1.3326260945
1.2418393096
1.1639555465
1.0963296631
1.0370061228
0.9845060725
0.9376890448
0.8956603972
0.857707722
0.8232561429
0.7918362416
0.7630606246
0.736606518
0.7122026506
0.6896192332
0.6686602103
0.649157202
0.6309647172
0.6139563374
0.5980216466
0.5830637406
0.5689971932
0.5557463819
0.5432441017
0.5314304096
0.5202516569
0.5096596728
0.4996110744
Z vs w Z1=1+i2w, Z2=1-4i/wResonance at 1.414
Sheet1
Test of COMComplex
abFormatMagrecipsummult
343.00 + i(4.00)50.1200 + i(0.1600)16.0000 + i(63.0000)
12512.00 + i(5.00)130.0710 + i(0.0296)
Sheet2
wALBCZwMag
0.2001.0000.4001.000-20.0001.0360.3530.2001.095
0.4001.0000.8001.000-10.0001.1580.7270.4001.367
0.6001.0001.2001.000-6.6671.4131.1290.6001.809
0.8001.0001.6001.000-5.0001.9001.5300.8002.439
1.0001.0002.0001.000-4.0002.7501.7501.0003.260
1.2001.0002.4001.000-3.3333.8741.3411.2004.100
1.4001.0002.8001.000-2.8574.4970.1001.4004.498
1.6001.0003.2001.000-2.5004.118-1.0911.6004.260
1.8001.0003.6001.000-2.2223.374-1.6351.8003.749
2.0001.0004.0001.000-2.0002.750-1.7502.0003.260
2.2001.0004.4001.000-1.8182.313-1.6942.2002.867
2.4001.0004.8001.000-1.6672.013-1.5872.4002.564
2.6001.0005.2001.000-1.5381.804-1.4722.6002.329
2.8001.0005.6001.000-1.4291.654-1.3642.8002.144
3.0001.0006.0001.000-1.3331.543-1.2673.0001.997
3.2001.0006.4001.000-1.2501.459-1.1813.2001.877
3.4001.0006.8001.000-1.1761.393-1.1053.4001.778
3.6001.0007.2001.000-1.1111.341-1.0383.6001.695
3.8001.0007.6001.000-1.0531.299-0.9783.8001.626
4.0001.0008.0001.000-1.0001.264-0.9254.0001.566
4.2001.0008.4001.000-0.9521.235-0.8774.2001.515
4.4001.0008.8001.000-0.9091.211-0.8344.4001.470
4.6001.0009.2001.000-0.8701.191-0.7944.6001.431
4.8001.0009.6001.000-0.8331.173-0.7594.8001.397
5.0001.00010.0001.000-0.8001.158-0.7275.0001.367
5.2001.00010.4001.000-0.7691.145-0.6975.2001.3
