The Convergence of Complex Integrals

In summary, for any a>0 and k,t\in\mathbb{R}, the integral \int_0^a t^k\; dt converges if and only if k>-1. Additionally, if k is a complex number, then the integral \int_0^a |t^k| \; dt also converges if and only if the real part of k is greater than -1. This is because for any real and positive value of t, the absolute value of t^k is equal to t^x, where x is the real part of k. Therefore, the convergence of the integral is dependent on the value of x, which must be greater than -1 for the integral to converge
  • #1
Ted123
446
0
I know that for any [itex]a>0[/itex] and [itex]k,t\in\mathbb{R}[/itex], the integral [tex]\int_0^a t^k\; dt[/tex] converges if and only if [itex]k>-1[/itex].

Is it true that if [itex]k[/itex] is complex then [tex]\displaystyle \int_0^a |t^k| \; dt[/tex] converges if and only if [itex]\text{Re}(k)>-1[/itex] since if [itex]t[/itex] is real, [itex]|t^k|[/itex] does not depend on the imaginary part of [itex]k[/itex]?
 
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  • #2
Ted123 said:
I know that for any [itex]a>0[/itex] and [itex]k,t\in\mathbb{R}[/itex], the integral [tex]\int_0^a t^k\; dt[/tex] converges if and only if [itex]k>-1[/itex].

Is it true that if [itex]k[/itex] is complex then [tex]\displaystyle \int_0^a |t^k| \; dt[/tex] converges if and only if [itex]\text{Re}(k)>-1[/itex] since if [itex]t[/itex] is real, [itex]|t^k|[/itex] does not depend on the imaginary part of [itex]k[/itex]?

Yes, it is. [itex]t^k=e^{log(t) k}[/itex]. If k is pure imaginary and t>0 |t^k|=1.
 
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  • #3
Dick said:
Yes, it is. [itex]t^k=e^{log(t) k}[/itex]. If k is pure imaginary and t>0 |t^k|=1.

So would you prove it as follows:

If [itex]x+iy=k\in\mathbb{C}[/itex] and [itex]t,x,y\in\mathbb{R}[/itex] with [itex]t>0[/itex] then [tex]|t^k| = |t^{x+iy}| = |t^x t^{iy}| = |t^x e^{\ln(t)iy}| = |t^x||e^{\ln(t)iy}| = |t^x| = t^x[/tex] so we're just back in the real case where we know [itex]\int_0^a t^x dt[/itex] converges for [itex]x=\text{Re}(k)>-1[/itex].

It is true that [itex]|t^x| = t^x[/itex] for any real [itex]x[/itex] and positive real [itex]t[/itex] isn't it?
 
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  • #4
Ted123 said:
So would you prove it as follows:

If [itex]x+iy=k\in\mathbb{C}[/itex] and [itex]t,x,y\in\mathbb{R}[/itex] with [itex]t>0[/itex] then [tex]|t^k| = |t^{x+iy}| = |t^x t^{iy}| = |t^x e^{\ln(t)iy}| = |t^x||e^{\ln(t)iy}| = |t^x| = t^x[/tex] so we're just back in the real case where we know [itex]\int_0^a t^x dt[/itex] converges for [itex]x=\text{Re}(k)>-1[/itex].

It is true that [itex]|t^x| = t^x[/itex] for any real [itex]x[/itex] and positive real [itex]t[/itex] isn't it?

Sure. Why would you doubt that?
 

What is the definition of convergence of an integral?

The convergence of an integral refers to the property of a definite integral where the limit of the integral approaches a finite value as the boundaries of the integral approach infinity or some other defined limit.

How do you determine if an integral is convergent or divergent?

To determine if an integral is convergent or divergent, one can use various tests such as the comparison test, limit comparison test, or the integral test. These tests involve analyzing the behavior of the integrand and the limits of the integral to determine if it approaches a finite value or diverges to infinity.

What is the significance of convergence of an integral in mathematics?

The convergence of an integral is significant in mathematics as it allows for the evaluation of otherwise difficult or impossible integrals. It also plays a crucial role in the development of many mathematical concepts and theories, such as the Fundamental Theorem of Calculus and Fourier series.

Can an integral be both convergent and divergent?

No, an integral cannot be both convergent and divergent. It can only have one of these properties. If an integral is convergent, it means that the limit of the integral approaches a finite value. If it is divergent, it means that the limit does not exist or approaches infinity.

What are some real-life applications of convergence of an integral?

The convergence of an integral has many real-life applications in fields such as physics, engineering, and economics. For example, it is used in calculating the area under a curve, finding the center of mass of an object, and determining the total cost or profit in economic models.

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