by Problem 16.13. in .NET

Generating QR Code 2d barcode in .NET by Problem 16.13.

by Problem 16.13.
Recognizing QR-Code In .NET
Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in Visual Studio .NET applications.
Encode Quick Response Code In .NET
Using Barcode creator for Visual Studio .NET Control to generate, create QR Code ISO/IEC18004 image in .NET applications.
SERIES AND SINGULARITIES 16.18. For what values of z does each series converge
QR Code Recognizer In VS .NET
Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications.
Paint Barcode In .NET Framework
Using Barcode creator for .NET framework Control to generate, create bar code image in VS .NET applications.
1 X zn : n2 2n n 1
Barcode Scanner In .NET
Using Barcode reader for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications.
QR Code JIS X 0510 Creation In Visual C#.NET
Using Barcode generator for VS .NET Control to generate, create QR image in Visual Studio .NET applications.
The nth term un
Make QR-Code In Visual Studio .NET
Using Barcode creation for ASP.NET Control to generate, create QR image in ASP.NET applications.
QR Code Generation In Visual Basic .NET
Using Barcode encoder for .NET Control to generate, create QR Code ISO/IEC18004 image in .NET framework applications.
zn : Then n 2n
Creating UCC-128 In VS .NET
Using Barcode printer for .NET Control to generate, create GS1-128 image in .NET applications.
Create UPC-A Supplement 5 In .NET
Using Barcode encoder for VS .NET Control to generate, create Universal Product Code version A image in Visual Studio .NET applications.
     un 1  zn 1 n2 2n  jzj    lim  n  lim  n!1 un  n!1 n 1 2 2n 1 z  2 By the ratio test the series converges if jzj < 2and diverges if jzj > 2. If jzj 2 the ratio test fails. 1 1 1 X1 X zn  X jzjn    converges if jzj 2, since However, the series of absolute values n2 2n  n2 2n n2 n 1 n 1 n 1 converges. Thus, the series converges (absolutely) for jzj @ 2, i.e., at all points inside and on the circle jzj 2. b
Draw Barcode In .NET
Using Barcode creator for VS .NET Control to generate, create barcode image in .NET applications.
Drawing UPCE In .NET Framework
Using Barcode generation for .NET framework Control to generate, create UPCE image in .NET applications.
1 X 1 n 1 z2n 1 n 1
Creating Code 3/9 In None
Using Barcode encoder for Font Control to generate, create Code 39 image in Font applications.
Recognizing Code-39 In Java
Using Barcode decoder for Java Control to read, scan read, scan image in Java applications.
2n 1 !
Generating Code128 In Java
Using Barcode drawer for Java Control to generate, create ANSI/AIM Code 128 image in Java applications.
Bar Code Creator In None
Using Barcode printer for Microsoft Word Control to generate, create barcode image in Office Word applications.
z3 z5 : 3! 5!
Print UPC - 13 In Visual Studio .NET
Using Barcode generation for ASP.NET Control to generate, create EAN 13 image in ASP.NET applications.
GS1-128 Generation In None
Using Barcode creation for Font Control to generate, create UCC.EAN - 128 image in Font applications.
We have
Bar Code Creation In Java
Using Barcode maker for Android Control to generate, create bar code image in Android applications.
EAN-13 Supplement 5 Reader In Visual C#.NET
Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications.
        z2  n 2n 1 un 1  2n 1 !       lim  1 z lim  lim   0  n!1 un  n!1 2n 1 ! 1 n 1 z2n 1  n!12n 2n 1  Then the series, which represents sin z, converges for all values of z.
CHAP. 16]
FUNCTIONS OF A COMPLEX VARIABLE     n 1  un 1  3n  jz ij    lim  z i : We have lim    n!1 un  n!1 3 z i n  3n 1
1 X z i n n 1
The series converges if jz ij < 3, and diverges if jz ij > 3. 1 X in If jz ij 3, then z i 3ei and the series becomes e .
This series diverges since the nth
term does not approach zero as n ! 1. Thus, the series converges within the circle jz ij 3 but not on the boundary.
1 X n 0
16.19. If
an zn is absolutely convergent for jzj @ R, show that it is uniformly convergent for these
values of z.
The de nitions, theorems, and proofs for series of complex numbers are analogous to those for real series. 1 X Mn converges, it follows by the In this case we have jan zn j @ jan jRn Mn . Since by hypothesis 1 n 1 X n an z converges uniformly for jzj @ R. Weierstrass M test that
16.20. Locate in the nite z plane all the singularities, if any, of each function and name them.
a z2 : z 1 3 z 1 is a pole of order 3.
2z3 z 1 . z 4 z i z 1 2i order 1 (simple poles).
z 4 is a pole of order 2 (double pole); z i and z 1 2i are poles of
p sin mz 2 4 8 2 2i 1 i, we can write , m 6 0. Since z2 2z 2 0 when z 2 2 z2 2z 2 2 z 2z 2 fz 1 i gfz 1 i g z 1 i z 1 i . The function has the two simple poles: z 1 i and z 1 i. 1 cos z . z 0 appears to be a singularity. z singularity. However, since lim 1 cos z 0, it is a removable z
Another method: ( !) 1 cos z 1 z2 z4 z6 z z3 1 1 Since , we see that z 0 is a removaz z 2! 4! 2! 4! 6! ble singularity. e e 1= x 1 1
1 1 : z 1 2 2! z 1 4
This is a Laurent series where the principal part has an in nite number of non-zero terms. Then z 1 is an essential singularity. ( f ) ez . This function has no nite singularity. However, letting z 1=u, we obtain e1=u , which has an essential singularity at u 0. We conclude that z 1 is an essential singularity of ez . In general, to determine the nature of a possible singularity of f z at z 1, we let z 1=u and then examine the behavior of the new function at u 0.
16.21. If f z is analytic at all points inside and on a circle of radius R with center at a, and if a h is any point inside C, prove Taylor s theorem that
Copyright © OnBarcode.com . All rights reserved.