What-is-the-meaning-of-i-z-0-ii-z-z-0-z-z-0-C-

Question Number 2874 by RasheedAhmad last updated on 29/Nov/15

W h a t i s t h e m e a n i n g o f

(i) z \to 0 (i i) z \to z_{0} z, z_{0} \in C

Answered by Filup last updated on 29/Nov/15

(1)

z approaches 0

z \to 0^{+} approaches from right side

z \to 0^{-} approaches from left side

e . g .

f (x) = {\begin{cases} x ⩾ 0, 1 \\ x < 0, 2 \end{cases}

\lim_{x \to 0^{+}} f (x) = 1

\lim_{x \to 0^{-}} f (x) = 2

(2)

z, z_{0} \in C means that both z and z_{0}

are within the complex set . Simply,

they are complex functions .

Such as : z = 1 + 2 i

if z \to z_{0}, then for complex z, it

approaches complex z_{0}

Answered by Yozzi last updated on 29/Nov/15

F o r z \in C d e f i n e d s i m p l y a s z = x + i y

w h e r e x, y \in R,

(i) z \to 0 \Rightarrow x + i y \to 0 \Rightarrow (x \to 0) \land (y \to 0)

(i i) z \to z_{0} \Rightarrow x + i y \to x_{0} + {i y}_{0}

\Rightarrow (x \to x_{0}) \land (y \to y_{0}) .

E . g D e f i n e f (z) = z^{2}, z \in C .

\Rightarrow \lim_{z \to 0} f (z) = \lim_{z \to 0} z^{2} = 0^{2} = 0

B u t, i f z = x + i y, x, y \in R,

f (x) = {(x + i y)}^{2} = x^{2} - y^{2} + 2 i y x

∴ \lim_{z \to 0} f (z) = \lim_{(x, y) \to (0, 0)} (x^{2} - y^{2} + 2 i x y)

= 0^{2} - 0^{2} + 2 i \times 0 \times 0

= 0

S i m i l a r l y, s u p p o s e z_{0} = 3 + i .

\lim_{z \to z_{0}} f (z) = \lim_{z \to z_{0}} z^{2} = {(3 + i)}^{2} = 8 + 6 i

A l t e r n a t i v e l y,

\lim_{z \to z_{0}} f (z) = \lim_{(x, y) \to (3, 1)} x^{2} - y^{2} + 2 i x y

= 9 - 1 + 2 i \times 1 \times 3

= 8 + 6 i

Commented by Yozzi last updated on 29/Nov/15

I s e e . T h a n k s!

Commented by 123456 last updated on 29/Nov/15

however there infinite ways to aproach

z_{0} on C, so if the limit depends on path

that you taked, them it doenst exist

[like in the R, \lim_{x \to 0^{-}} f (x) \neq \lim_{x \to 0^{+}} f (x), \lim_{x \to 0} f (x) = ∄]

Commented by prakash jain last updated on 29/Nov/15

The same concept (path dependency applies)

when computing limits for functions of

more than one variable .

for z \to 0 or z \to z_{0} limits it is sometimes easier to

convert to polar coordinates then consider

limit on r with no constraints on θ .

Commented by Rasheed Soomro last updated on 29/Nov/15

W h a t a r e o t h e r w a y s, f o r e x a m p l e, o f a p p r o a c h i n g z_{0}

a n d w h a t i s^{'} p a t h {d e p e n d a n c y}^{'} ?

Commented by 123456 last updated on 29/Nov/15

lets z_{0} = 0

we can aproach z b y

z = {t e}^{ı t}, t \to 0^{+} (path 1)

z = {r e}^{ı θ}, r \to 0 (path 2)

the first path is a spiral and the second

is a stray line with angle θ with real line

path depedency = limits depends on the

path you take to aproach z_{0}

in real case with one varriable we have

only the paths

x \to 0^{+}

x \to 0^{-}

however in C we have infinites path

as you see in above a example of two

path, there are more paths that you

can take .

Commented by Rasheed Soomro last updated on 29/Nov/15

T {HANK}^{S s s} f o r a l l o f Y OU!

I a l w a y s h a v e l e a r n t f r o m y o u .