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Zeros and Poles | Removable Singularity | Complex Analysis #7
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Everything you need to know about Zeros, Poles and Removable Singularity. The video also includes a lot of examples for each concept.
LINK TO COMPLEX ANALYSIS PLAYLIST
LINK TO CANVAS
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Consider subscribing, liking or leaving a comment, if you enjoyed the video or if it helped you understand the subject. It really helps me a lot.
IMPORTANT LINKS
CONCEPTS FROM THE VIDEO
► Zeros
Also known as "roots" to an analytic function. A point z_0 in the complex plane is a zero of order n for some analytic function f(z) if:
f(z_0) = f'(z_0) = f''(z_0) = ... = f^(n-1) (z_0) = 0 and f^(n) (z_0) =\= 0.
An alternative definition is that f(z) has a zero of order n at z_0 only if it can be rewritten as
f(z) = (z-z_0)^n * g(z), where g(z_0) =\= 0
► Poles
A pole is a specific kind of singularity of a complex function, it behaves as the singularity of 1/z^n at z = 0. So the most intuitive definition is that poles are points z_0 in the complex plane so that f(z_0) = g(z_0)/0, where g(z_0) =\= 0.
But the more formal definition is that a point z_0 in the complex plane is a pole of order n for some analytic function f(z) if:
lim_(as z goes to z_0) of (z - z_0)^n * f(z) =\= 0
A pole of order one is called a simple pole and if we only observe the neighborhood around z_0 when the relation above take the following form
f(z) = g(z) / (z-z_0)^n, where g(z_0) =\= 0
► Removable Singularity
A removable singularity is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighborhood of that point.
EDITS:
The 2:nd derivative at 03:20 is wrong, it should be g''(z) = 2, however does not change the order of the zero since g''(-4) =/= 0.
Missed "-" at 04:00, since the derivative of cos(z) is -sin(z).
TIMESTAMPS
00:00 - 00:11 Intro
00:11 - 00:52 Definition Zeros
00:52 - 02:40 Definition Poles
Example Zeros:
1) z-1 02:40 - 03:09
2) (z+4)^2 03:09 - 03:51
3) cos(z*pi/2) 03:51 - 04:13
4) (z-1)cos(z*pi/2) 04:13 - 04:40
Example Poles:
1) 1/(z-1) 04:40 - 05:35
2) 2/(z+3)^2 05:35 - 06:39
Zero and Pole at the same point: 06:39 - 07:51
Definition Removable Singularity: 07:51 - 08:15
Example Removable Singularity:
1) ((z-1)(z+2))/((z-1)(z+3)^2(z+1)) 08:15 - 08:55
2) sin(z)/z^3 08:55 - 10:04
SOCIAL
HASHTAGS
#TheMathCoach #ComplexAnalysis
LINK TO COMPLEX ANALYSIS PLAYLIST
LINK TO CANVAS
SUPPORT
Consider subscribing, liking or leaving a comment, if you enjoyed the video or if it helped you understand the subject. It really helps me a lot.
IMPORTANT LINKS
CONCEPTS FROM THE VIDEO
► Zeros
Also known as "roots" to an analytic function. A point z_0 in the complex plane is a zero of order n for some analytic function f(z) if:
f(z_0) = f'(z_0) = f''(z_0) = ... = f^(n-1) (z_0) = 0 and f^(n) (z_0) =\= 0.
An alternative definition is that f(z) has a zero of order n at z_0 only if it can be rewritten as
f(z) = (z-z_0)^n * g(z), where g(z_0) =\= 0
► Poles
A pole is a specific kind of singularity of a complex function, it behaves as the singularity of 1/z^n at z = 0. So the most intuitive definition is that poles are points z_0 in the complex plane so that f(z_0) = g(z_0)/0, where g(z_0) =\= 0.
But the more formal definition is that a point z_0 in the complex plane is a pole of order n for some analytic function f(z) if:
lim_(as z goes to z_0) of (z - z_0)^n * f(z) =\= 0
A pole of order one is called a simple pole and if we only observe the neighborhood around z_0 when the relation above take the following form
f(z) = g(z) / (z-z_0)^n, where g(z_0) =\= 0
► Removable Singularity
A removable singularity is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighborhood of that point.
EDITS:
The 2:nd derivative at 03:20 is wrong, it should be g''(z) = 2, however does not change the order of the zero since g''(-4) =/= 0.
Missed "-" at 04:00, since the derivative of cos(z) is -sin(z).
TIMESTAMPS
00:00 - 00:11 Intro
00:11 - 00:52 Definition Zeros
00:52 - 02:40 Definition Poles
Example Zeros:
1) z-1 02:40 - 03:09
2) (z+4)^2 03:09 - 03:51
3) cos(z*pi/2) 03:51 - 04:13
4) (z-1)cos(z*pi/2) 04:13 - 04:40
Example Poles:
1) 1/(z-1) 04:40 - 05:35
2) 2/(z+3)^2 05:35 - 06:39
Zero and Pole at the same point: 06:39 - 07:51
Definition Removable Singularity: 07:51 - 08:15
Example Removable Singularity:
1) ((z-1)(z+2))/((z-1)(z+3)^2(z+1)) 08:15 - 08:55
2) sin(z)/z^3 08:55 - 10:04
SOCIAL
HASHTAGS
#TheMathCoach #ComplexAnalysis
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