Problem with roots of a function h(x)

In summary, the conversation discusses the function h(x)=f(x)+g(x) and how to find the values of x where h(x)=0. It is noted that there are two types of roots for this function, one where f(x)=-g(x) and another where both f(x) and g(x) equal 0. The conversation then delves into the topic of proving that the roots of f(x) and g(x) also satisfy the formula f(a)/g(a)=-1 with a being a root of both functions. However, this claim is ultimately proven to be false in general. The conversation also touches on the function h(x)=F(a+ix)e^{-iwx}+F(a-ix)e^{
  • #1
eljose
492
0
Let be the function h(x)=f(x)+g(x) we want to obtain the values of x so h(x)=0 and we have that f(x) and g(x) have the same roots (if a is so f(a)=0 then g(a)=0 too) so we have two types of roots:

1.-values of x that satisfy f(x)=-g(x)
2.values x that make f(x)=g(x)=0

then we make in the first equation f(x)/g(x)=-1 this should be valid for any root of h(x) the problem would be if the roots of f(x) and g(x) would also satisfy the formula f(a)/g(a)=-1 with a a root of f(x) and g(x) ,we have that the values that satisfy the equation h(x)=0->that f(x)/g(x)=-1 so as a is a root of f and g would also satisfy that h(a)=0=0+0 and so should satisfy that f(x)/g(x)=-1 but how to prove this?...thanks.
 
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  • #2
If a is a root of both f and g, f(a)/g(a) is not equal to -1. It is 0/0, which is meaningless. It may or may not be the case that [tex]\lim_{x\rightarrow a} \frac{f(x)}{g(x)}=-1[/tex]
 
  • #3
I'm having a very difficult time trying to parse what you're saying.


It almost sounds as if you're suggesting that, if a is a root of h, then it is always true that f(a) / g(a) = -1, but that's clearly false, by the very observation you made, that it might be the case that f(a) = g(a) = 0.
 
  • #4
but we may have that the limit Lim x->a [f(x)/g(x)]=1 with []=modulus of the function.

another question,let,s suppose we have the function h(x)=F(a+ix)e^{-iwx}+F(a-ix)e^{iwx} so the function is real,now we want to obtain the roots of h(x)=0 F(x) is real so F(a+ix) and F(a-ix) are one the conjugate of other let,s suppose c is a root of both F(a+ix) and F(a-ix) so F(a+ic)=F(a-ic)=0 then would we have that the limit when x-> [F(a+ix)/F(a-ix)]=1..

my reasioning is that expanding F(a+ix) and F(a-ix) near the root c and supposing F(a+ix) and F(a-ix) don,t have an extremum at x=c we have that:

F(a+ix)=F`(a+ic)i(x-c) and F(a-ix)=F´(a-ic)(x-c)(-i) so dividing one by other we would have that Limx->c F(a+ix)/F(a-ix)=(-1)F`(a+ic)/F´(a-ic) but we also have that ^F´(a+ic) and F`(a-ic) are conjugate so their quotient will be e^{id} for some real d but taking the modulus of [F´(a+ic)/F´(a-ic)(-1)]=1 as we wanted to prove...
 
  • #5
Please use correct punctuation. It greatly helps readability. :tongue2: e.g. periods at ends of sentences, and writing apostrophes as apostrophes. (It's "let's", not "let,s")


If h is given by [itex]h(x)=F(a+ix)e^{-iwx}+F(a-ix)e^{iwx}[/itex], that is not sufficient to conclude that h is real.

If b is a root of h, that does not imply that F(a + ib) = 0.

Also, when z is nonzero, we have the identity [itex]|z / \bar{z}| = 1[/itex].
 
  • #6
thanks for your reply, but if F(x) is real then F(a+ix) and F(a-ix) (if we set w=0) then F(a+ix) would be the complex conjugate of F(a-ix) and viceversa (expanding both functions in x would give the answer)...

if b is a root of h this doesn,t not imply that F(a+ib)=0 but you will agree that the set of roots that satisfy F(a+ib)=0 is included in the set of roots of h(x) (if b is a root of F(a+ix) then is also a root of h(x)=0)
 
  • #7
eljose said:
thanks for your reply, but if F(x) is real then F(a+ix) and F(a-ix) (if we set w=0) then F(a+ix) would be the complex conjugate of F(a-ix) and viceversa (expanding both functions in x would give the answer)...
pleaes provide a proof of this statement. hint, it is false, in general, and you've not said what F, x, or a is. I mean is x some particular nubmer, real or complex, or are you attemping to say "for all x"? never mind if that is for x real or complex.

expanding in "x" makes no sense. Who said that F was analytic? not you.
 
  • #8
but you will have that f(a+it)=R(t)e^{iv(t)} and f(a-it)=R(t)e^{-iv(t)} with t real and v(t) takes real values...

Edit:well i think that the f(a+it)+f(a-it) will only be real if [tex] f^{*}(z)=f(z^*) [/tex] with *=complex conjugate...

tehn if f(a+it)+f(a-it) is real then will be true that f(a+it)?R(t)exp{iv(t)} and
f(a-it)=R(t)exp{-iv(t)} so the sum will be real...

then if f(a+it)+f(a-it)=z(t) is real then its derivatives D^{n}Z(t) will also be real so we will be able to express D^{n}Z(t)=W_n(t)exp{iU_n(t)} for the sets of function U_n(t) and W_n(t).
 
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  • #9
you' ve not answered any of the questions asked of you. Let us show you by example:

Let F(z)=i if im(z)=/=0 and 1 if z is purely real.

then for any real x, F(x) is real but F(a+ix) and F(a-ix) do not satisfy your claim of complex conjugacy nor of h(x) being a real valued function, indeed h(2pi)=2i

why don't you take care and write things out properly explaining all the assumptions you are making? What, for instance is R(t), and v(t) now? not to mention all the other undefined terms you've now added to your edited post
 
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  • #10
eljose said:
Edit:well i think that the f(a+it)+f(a-it) will only be real if [tex] f^{*}(z)=f(z^*) [/tex] with *=complex conjugate...


but you didn't specify this requirement originally, did you? and it isn't even true ofr f analytic. it would be true if, say there were a taylor series convergent everywhere and with purely real coefficients.
 
  • #11
and let be the f with f real that have f(a+ib)=exp(ik) and f(c+id)=exp(-ik) then should we have necessaary that a=c and b=-d (k is the same for both cases)...
 
  • #12
what on Earth are you on about now? where did the k come from? f is now a function from C to R?... madness
 
  • #13
k is a real constant we have that f(a+ib) and f(c+id) have the same modulus...then my reasoning is that if we express f(a+ib) and f(c+id) in terms of their real and complex part we would have that f(a+ib)=U(a,b)+iV(a,b) and f(c+id)=U(c,d)+iV(c,d) then as their modulus is 1 then we have that U(a,b)=U(c,d) and V(a,b)=V(c,d)

we have also that their complex part is the same but a sign so we would have that...
Artan[V(a,b)/U(a,b)]=-Artan[V(c,d)/U(c,d)] from this i think we could deduce that a=c
 
  • #14
if this reflects your standard of presentation in these things you write then no wonder they are rejected. even now no one is any the wiser as to what you mean. none of what you have written can with any reasonable certainty be said to ve true, it is almost surely false. think it's time to stop wasting time in working out what you're trying to say. try again when you can make it readable.
 
  • #15
i have another question let be [tex] [f(a+ib)]^* [/tex] the complex conjugate of the function [tex] f(a+ib) [/tex] then my question is if we denote f^{-1} the inverse function of f i would like to know if the equality holds...

[tex] f^{-1}*(f^*(a+ib))=a+ib [/tex] that is the conjugate of the inverse function applied to the conjugate function is the x function...
 
  • #16
why are youi asking? you don't bother to listen to any replies.
 

What is the definition of "roots of a function"?

The roots of a function h(x) are the values of x that make the function equal to zero. In other words, these are the values of x that satisfy the equation h(x) = 0.

How do you find the roots of a function algebraically?

To find the roots of a function algebraically, you can use techniques such as factoring, the quadratic formula, or completing the square. These methods involve manipulating the function algebraically until it is in the form h(x) = 0 and then solving for the value(s) of x.

Can a function have more than one root?

Yes, a function can have multiple roots. This happens when there are multiple values of x that make the function equal to zero. For example, the function h(x) = x^2 - 4 has two roots, x = 2 and x = -2.

What is the significance of the roots of a function?

The roots of a function are important because they represent the points where the function intersects the x-axis. This can provide valuable information about the behavior of the function and can be used to solve equations and real-world problems.

Can a function have imaginary roots?

Yes, a function can have imaginary roots. This happens when the values of x that make the function equal to zero involve complex numbers. For example, the function h(x) = x^2 + 4 has two imaginary roots, x = 2i and x = -2i.

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