Understanding Hyperbolic Functions: A Worked Example from Definitions

In summary, the conversation is about working from definitions and a specific example from a book. The person struggles with a step that involves getting all the variables to be positive, and also notes that when plugging in numbers, the two equations do not equal each other. The expert suggests multiplying the second equation by e^u to find the values of u for which the equation is satisfied.
  • #1
Firepanda
430
0
Cosh u = (2sinh u) -1

Working from definitions

http://img118.imageshack.us/img118/3271/eusm4.png

Its a worked example from the book, which isn't very well explained. The only step i struggle on is from how the managed to get all the u's positive (step 2). I plugged some numbers in for u and find the two equations don't equal each other anyway. What am i doing wrong?

thx
 
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  • #2
Firepanda said:
Cosh u = (2sinh u) -1

The only step i struggle on is from how the managed to get all the u's positive (step 2).

What happens when you multiply the second equation in your linked image by e^u?

I plugged some numbers in for u and find the two equations don't equal each other anyway. What am i doing wrong?

Right, this equation isn't satisfied by all values of u. You are supposed to find the values of u for which it is satisfied.
 
  • #3
George Jones said:
What happens when you multiply the second equation in your linked image by e^u?



Right, this equation isn't satisfied by all values of u. You are supposed to find the values of u for which it is satisfied.

gotcha thx!
 

Related to Understanding Hyperbolic Functions: A Worked Example from Definitions

1. What are simple hyperbolic functions?

Simple hyperbolic functions are mathematical functions that involve the hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh) of a given number. They are derived from the exponential function and share many properties with trigonometric functions.

2. What is the difference between simple and complex hyperbolic functions?

Simple hyperbolic functions only involve the basic hyperbolic functions (sinh, cosh, tanh), while complex hyperbolic functions involve other operations such as exponentiation and logarithms. Simple hyperbolic functions can be expressed in terms of complex hyperbolic functions, but not vice versa.

3. How are simple hyperbolic functions used in real life?

Simple hyperbolic functions have many applications in physics, engineering, and other sciences. They are used to model oscillatory systems, describe electric and magnetic fields, and solve differential equations, among other things. They also have practical uses in fields such as acoustics, optics, and signal processing.

4. What is the relationship between simple hyperbolic functions and trigonometric functions?

Simple hyperbolic functions are closely related to trigonometric functions. In fact, they are defined in terms of exponential functions, which are also related to trigonometric functions. The hyperbolic sine and cosine functions, for example, can be expressed in terms of the sine and cosine functions, respectively.

5. Are there any special properties of simple hyperbolic functions?

Yes, simple hyperbolic functions have several important properties that make them useful in mathematical analysis. They are odd and even functions, meaning they have symmetry about the origin. They also have derivatives and integrals that can be easily calculated, and they satisfy a variety of identities and equations, making them valuable tools in solving mathematical problems.

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