vba code for barcode in excel Residue Theory in Java

Drawer Quick Response Code in Java Residue Theory

Residue Theory
QR Code 2d Barcode Printer In Java
Using Barcode printer for Java Control to generate, create QR Code JIS X 0510 image in Java applications.
Denso QR Bar Code Decoder In Java
Using Barcode decoder for Java Control to read, scan read, scan image in Java applications.
The most general solution is a superposition of such solutions which ranges over all possible values of n Therefore we write u (r , ) = r n (an cos n + bn sin n ) = a0 + r n (an cos n + bn sin n )
Barcode Creation In Java
Using Barcode creator for Java Control to generate, create bar code image in Java applications.
Decode Bar Code In Java
Using Barcode scanner for Java Control to read, scan read, scan image in Java applications.
n=0 n =1
Encoding QR Code 2d Barcode In C#
Using Barcode generation for Visual Studio .NET Control to generate, create QR image in .NET applications.
QR Code JIS X 0510 Maker In .NET
Using Barcode printer for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications.
To proceed, the following orthogonality integrals are useful:
Make QR In VS .NET
Using Barcode creation for .NET framework Control to generate, create Quick Response Code image in .NET applications.
Denso QR Bar Code Printer In Visual Basic .NET
Using Barcode encoder for .NET framework Control to generate, create QR Code ISO/IEC18004 image in VS .NET applications.
2 0
Making USS-128 In Java
Using Barcode generator for Java Control to generate, create UCC - 12 image in Java applications.
Paint Code-39 In Java
Using Barcode generation for Java Control to generate, create Code 39 image in Java applications.
mn sin m sin n d = 0
Drawing Data Matrix 2d Barcode In Java
Using Barcode drawer for Java Control to generate, create Data Matrix image in Java applications.
Generating Linear 1D Barcode In Java
Using Barcode drawer for Java Control to generate, create Linear image in Java applications.
for n 0 for n = 0 for n 0 for n = 0
Encode International Standard Serial Number In Java
Using Barcode maker for Java Control to generate, create ISSN image in Java applications.
GTIN - 128 Printer In Objective-C
Using Barcode maker for iPad Control to generate, create EAN / UCC - 13 image in iPad applications.
(78)
ANSI/AIM Code 128 Recognizer In .NET
Using Barcode recognizer for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications.
UPC-A Supplement 5 Scanner In .NET Framework
Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications.
2 0
Data Matrix 2d Barcode Maker In .NET Framework
Using Barcode generation for ASP.NET Control to generate, create Data Matrix 2d barcode image in ASP.NET applications.
UPC-A Supplement 5 Encoder In Objective-C
Using Barcode generation for iPhone Control to generate, create UPC-A image in iPhone applications.
mn cos m cos n d = 2 mn sin m cos n d = 0
Barcode Generator In None
Using Barcode generation for Word Control to generate, create bar code image in Word applications.
Printing UCC.EAN - 128 In None
Using Barcode generation for Online Control to generate, create EAN / UCC - 13 image in Online applications.
(79)
2 0
(710)
Here, mn = 1 for m = n, 0 which is the Kronecker delta function Now we apply the boundary condition u (1, ) = f ( ) for 0 2 : f ( ) = a0 + r n (an cos n + bn sin n )
n =1
(711)
Multiply through this expression by sin m and integrate We obtain
2 0
f ( )sin m d = a0 sin m d
n =1
2 0
an cos n sin m d + bn sin n sin m d
n =1
bn sin n sin m d = bn mn = bm
n =1
where Eqs (78) (710) were used We conclude that bn = 1
2 0
f ( )sin n d
Complex Variables Demysti ed
Now we return to Eq (711), and multiply by cos m and integrate This time
2 0
f ( ) cos m d = a0 cos m d
n =1
2 0
an cos n cos m d + bn sin n cos m d
n =1
an sin n cos m d = an mn = am
n =1
Hence am = 1
2 0
f ( ) cos m d
To obtain the constant a0 , we integrate without rst multiplying by any trig functions, that is:
2 0
f ( ) d = a0 d +
0 n =1
2 0
an cos n d + bn sin n d
= a0 2 a0 = 1 2
2 0
f ( ) d
This should be obvious since
2 0 2 0
cos n d =
1 sin n n
1 sin n d = cos n n
Now we are in a position to derive Poisson s formula We have u (r , ) = 1 2
2 0
f ( ) d
1 + r n n =1
2 0
f ( ) cos n d cos n +
2 0
f ( )sin n d sin n
Residue Theory
We can move the summation inside the integrals: u (r , ) = 1 2 + = = 1 2 1 2
2 0
f ( ) d +
2 0
f ( )
1 n r cos n cos n d n =1
f ( )
1 n r sin n sin n d n =1
0 2 0
d f ( ) 1 + 2 r n cos n cos n + 2 r n sin n sin n n =1 n =1 d f ( ) 1 + 2 r n (cos n cos n + sin n sin n ) n =1
Now recall that cos n cos n + sin n sin n = cos(n( )) It s also true that 1 2 r n cos [ n( ) ] =
n =1
1 r2 1 2r cos( ) + r 2
(712)
So, we arrive at the Poisson formula for a disc of radius one: u (r , ) = 1 2
2 0
1 r2 f ( ) d 1 2r cos( ) + r 2
This tells us that the value of a harmonic function at a point inside the circle is the average of the boundary values of the circle
The Cauchy s Integral Formula as a Sampling Function
The Dirac delta function has two important properties First if we integrate over the entire real line then the result is unity:
( x ) dx = 1
Complex Variables Demysti ed
Second, it acts as a sampling function that is, it picks out the value of a real function f (x) at a point:
f ( x ) ( x a)dx = f (a)
In complex analysis, the function 1 / z plays an analogous role It has a singularity at z = 0, and 1 2 i
dz 0 = z 1
if 0 is not in the interior of if 0 is in the interior of
It also acts as a sampling function for analytic functions f (z) in that f (a) = 1 2 i
f ( z)dz z a
Some Properties of Analytic Functions
Now we are going to lay some more groundwork before we state the residue theorem In this section, we consider some properties of analytic functions
AN ANALYTIC FUNCTION HAS A LOCAL POWER SERIES EXPANSION
Suppose that a function f (z) is analytic inside a disc centered at a point a of radius r: z a < r Then f (z) has a power series expansion given by f ( z) = an ( z a)n
n=0
(713)
The coef cients of the expansion can be calculated using the Cauchy s integral formula in Eq (72): an = f ( n ) (a) n! (714)
Residue Theory
INTEGRATION OF THE POWER SERIES EXPANSION GIVES ZERO
Note the following result:
( z a)
if m 1 0 dz = ln( z a) if m = 1
Hence
f (z) dz = a (z a) dz
since n is never equal to 1
A FUNCTION f(z) THAT IS ANALYTIC IN A PUNCTURED DISC HAS A LAURENT EXPANSION
Consider the punctured disc of radius r centered at the point a We denote this by writing 0 < z a < r If f (z) is analytic in this region, it is analytic inside the disc but not at the point a In this case, the function has a Laurent expansion: f ( z) =
n =
a ( z a)
(715)
As stated in Chap 5, we can classify the points at which the function blows up or goes to zero A removable singularity is a point a at which the function appears to be unde ned, but it can be shown by writing down the Laurent expansion that in fact the function is analytic at a In this case the Laurent expansion in Eq (715) assumes the form f ( z) = an ( z a)n
Copyright © OnBarcode.com . All rights reserved.