Integrating copmplex function (proof of one property)

**sedam7** · September 1st 2011, 05:53 PM

Hello

(sorry for bad usage of English)
i have some issue with this (got to much rusty on this). I need to proof that:

$\displaystyle \int c f(z) dz = c\int f(z) dz$

(I try to do it like this... so if I'm wrong or there's "smarter" way to do it i would be very grateful... perhaps i go way back to show what integral is... don't now... please help )

well...

Let the complex function $f(z) = u(x,y) + i v(x,y)$ is defined in region $G\subset \mathbb{C}$ and let the $l\in G$ be smooth oriented curve with starting point A and end point B. We make division like so:

$\pi : A = z_0, z_1, z_2, ..., z_{k-1}, z_k, ..., z_n = B$

and make than even finer division:

$\xi _k = \alpha _k + i\beta _k$ where (k = 1,2,3,...,n)

where point $\xi _k$ is point between points $z_{k-1}$ and $z_k$

we form integral sum for division $\pi$

$\displaystyle \varsigma _\pi = \sum_{k =1}^{n}\varphi (\xi _k )(z_k - z_{k-1})$

$\displaystyle \Delta z_k = z_k - z_{k-1}$

$\displaystyle \Delta z_k = \Delta x_k - \Delta y_k$

$\displaystyle f(\xi_k) = u(\alpha _k, \beta _k) + i v(\alpha _k, \beta_k)$

so we pot this all back in integral sum... and have:

$\displaystyle \varsigma _\pi = \sum_{k =1}^{n}\left [ u(\alpha _k, \beta _k)\Delta x_k - v(\alpha _k, \beta_k)\Delta y_k \right ] + i \sum_{k =1}^{n}\left [ u(\alpha _k, \beta _k)\Delta y_k + v(\alpha _k, \beta_k)\Delta x_k \right ]$

$\displaystyle \varsigma _\pi = \max_k |\Delta z_k|$

than we have definition...
"For complex number I we say it's Riemman's integral of the complex function f(z) on curve l from point A to point B if and only if :
$\displaystyle (\forall \varepsilon >0)(\exists \delta (\varepsilon )>0) \therefore |I-\varsigma _\pi | <\varepsilon$ for $\max_k |\Delta z_k|<\delta$

$\displaystyle \lim_{\varsigma _\pi \to \infty} \sum_{k=1}^{n}f(\xi _k)\Delta z_k = I$

if I exists than we can write...

$\displaystyle I = \int _l f(z) dz$

or

$\displaystyle I =(l) \int _A ^B f(z) dz$

so if this exists than if we have some complex function g(z) = c f(z) where c is constant than we can say that :

$\displaystyle \lim_{\varsigma _\pi \to \infty} \sum_{k=1}^{n}c f(\xi _k)\Delta z_k = I$

and since c is constant than c can come in front of sum and limit so ....

$\displaystyle c \lim_{\varsigma _\pi \to \infty} \sum_{k=1}^{n} f(\xi _k)\Delta z_k = I$

and we can say that we can write

$\displaystyle \int c f(z) dz = c\int f(z) dz$

I think this is way to simple to write this much (if it's true)... but I don't know and that's why i need help...

and one more thing... how to proof that :

$\displaystyle \int_{z_1} ^{z_2} A dz = A (z_2 - z_1)$

where A is constant, but without using Newton-Leibniz formula ?!

(it all seems to be trivial until i got start...

or I'm just complicating it for myself

)

Thanks for any help

**FernandoRevilla** · September 1st 2011, 11:18 PM

Originally Posted by sedam7

Hello

(sorry for bad usage of English)
i have some issue with this (got to much rusty on this). I need to proof that:

$\displaystyle \int c f(z) dz = c\int f(z) dz$

(I try to do it like this... so if I'm wrong or there's "smarter" way to do it i would be very grateful... perhaps i go way back to show what integral is... don't now... please help )

Could you transcribe the exact formulation of the question?. If we are talking only about indefinite integrals, the question is easier, use for example the definition: if $f(z),F(z)$ are analytic on a region $\mathcal{R}$ such that $F'(z)=f(z)$ , then $F(z)$ is called an indefinite integral or antiderivative of $f(z)$ denoted by $F(z)=\int f(z)dz$ . Now, apply a well known property about derivatives.

**sedam7** · September 2nd 2011, 12:07 AM

thanks

question goes like ::

if f(z) is integrable function, show that
$\int c f(z) dz = c \int f(z) dz$

there's nothing more to the question

(and yes this one is about indefinite integrals.... second proof is for finite integrals )

Originally Posted by FernandoRevilla

if $f(z),F(z)$ are analytic on a region $\mathcal{R}$ such that $F'(z)=f(z)$ , then $F(z)$ is called an indefinite integral or antiderivative of $f(z)$ denoted by $F(z)=\int f(z)dz$ . Now, apply a well known property about derivatives.

I don't seem to follow

(it looks like it's definition of the primitive function.. or just looks like to me

sorry

)

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