Complex fraction as a ratio of sines?

In summary, it is possible to write the complex fraction as a ratio of two sines by using hyperbolic sines and the standard definition of sin(z) in terms of the exponential. However, the arguments of the sines will be complex numbers. To simplify the exponent out front, one can use the hyperbolic sine function. Additionally, the complex fraction can be expressed as a linear equation of Re()+Im(). To do so, one can rationalize the denominator by multiplying both the numerator and denominator by the complex conjugate of the denominator.
  • #1
de1irious
20
0
Is it possible for me to write this complex fraction as a ratio of two sines? Thanks.

[tex]\frac{1-e^{101z}}{1-e^{z}}[/tex]
 
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  • #2
It is possible, but the arguments of the sines will be complex numbers.
 
  • #3
I think you could use hyperbolic sine but I don't know much about hyperbolic trig f(x)s yet
 
  • #4
That's fine, Mathman. If we allow sin to be evaluated at complex numbers, using the standard definition of sin(z) in terms of the exponential, how do we we rewrite it? Thanks
 
  • #5
1-ez=ez/2(e-z/2-ez/2)
=-2ez/2sin(z/2)
Similar expression for 1-e101z. Then take ratio.
 
  • #6
I don't think that works though because sin(z) puts the exponent of e as iz and -iz, not z. For instance, if I plug that into a calculator, it doesn't come out equal. I am trying to factor out the real part, but the ratio will not cancel to the point that I have just sines.
 
  • #7
Nvm, I just realized you were talking about hyperbolic sines. Thanks for the tip!

EDIT: However, now I get e[tex]^{50z}[/tex][tex]\frac{sinh101z/2}{sinhz/2}[/tex]

Any way to simplify the exponent out front?
 
Last edited:
  • #8
de1irious said:
I don't think that works though because sin(z) puts the exponent of e as iz and -iz, not z. For instance, if I plug that into a calculator, it doesn't come out equal. I am trying to factor out the real part, but the ratio will not cancel to the point that I have just sines.

Sorry, my bad. I forgot to put the i factor in for the sin. Using sinh is better.
 
  • #9
A related question I have is can we express the complex fraction as a linear equation of Re()+Im()?
 
  • #10
orstats said:
A related question I have is can we express the complex fraction as a linear equation of Re()+Im()?
That's pretty standard isn't it? You "rationalize" the denominator by mutltiplying both numerator and denominator by the complex conjugate of the denominator. Exactly what the computations are depends on the particular complex fraction. Do you mean "complex fraction" in the sens "fraction with complex numbers" or "fraction with fractions in the numerator and denominator"?

Can you give an example of what you had in mind?
 
  • #11
I am not sure what the complex conjugate would be for the denominator...

sin(az)
______
sin(bz)

=

exp(iaz) - exp(-iaz)
________________
exp(ibz) - exp(-ibz)

=?

Re() + Im()
 

Related to Complex fraction as a ratio of sines?

1. What is a complex fraction as a ratio of sines?

A complex fraction as a ratio of sines is a mathematical expression that involves fractions with trigonometric functions, specifically sine functions, in the numerator and denominator.

2. How is a complex fraction as a ratio of sines simplified?

To simplify a complex fraction as a ratio of sines, you can use trigonometric identities and properties to rewrite the fraction in a more simplified form. This may involve factoring, cancelling out common factors, and using the reciprocal or quotient identities.

3. What is the significance of complex fractions as ratios of sines in math?

Complex fractions as ratios of sines are important in many fields of math, including calculus, trigonometry, and physics. They are used to solve problems involving waves, oscillations, and periodic functions.

4. Can complex fractions as ratios of sines be converted into other forms?

Yes, complex fractions as ratios of sines can be converted into other forms, such as a single fraction with a trigonometric expression in the numerator and a constant in the denominator. They can also be written in terms of other trigonometric functions, such as cosine or tangent.

5. How can complex fractions as ratios of sines be applied in real-life situations?

Complex fractions as ratios of sines can be used to analyze and model real-life phenomena that involve periodic motion, such as the motion of a pendulum or a vibrating string. They are also used in engineering and design to calculate the amplitudes and frequencies of waves in different systems.

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