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# Prove some properties of the complex sine and cosine functions

The following definitions extend the sine and cosine functions to take arguments : Prove the following formulas, where are complex numbers and .

1. .
2. .
3. .
4. .
5. .
6. .

1. Proof. Using the given definition of the sine of complex numbers we have 2. Proof. Similar to part (a) we compute 3. Proof. We compute 4. Proof. The two computations are as follows, 5. Proof. We have, 6. Proof. We have, # Determine a differential equation governing a falling body in resisting medium

Modify the equations (Example 2 on page 314 of Apostol) for the velocity of a falling body in a resisting medium if the resistance of the medium is proportional to instead of to . Prove that the resulting differential equation can be written in each of the following forms: where . By integrating these find the following formulas for : where . Determine the value of as .

Starting with the equation in example 2 and modifying it so that the resistance is proportional to we have Using the chain rule as we did in the previous exercise we know Therefore, Letting we then have This is the first requested equation.

Alternatively, Integrating the first equation we find, Integrating the second equation, This implies # Compute the limit of the given function

Evaluate the limit. We recall the definition of the hyperbolic cosine in terms of the exponential, Using this we compute, # Find the limit of the given function

Find the value of the following limit. We will apply L’Hopital’s rule three times, Since this final limit exists, the chain of applications of L’Hopital’s was justified.

# Find the limit as x goes to 0 of (cosh x – cos x) / x2

Evaluate the limit. We use the definition of in terms of the exponential: and the expansions (page 287 of Apostol) of and as : Putting these together we evaluate the limit: # Find all x satisfying equations given in terms of sinh

Let be the number such that . Find all that satisfy the given equations.

1. .
2. .

1. We are given . From the formula for this means Then, from the given equation we have Thus, So, then we have Therefore we have 2. There can be no which satisfy the given equation. As in part (a), we use the definition of to obtain the equation, Next, we use the equation given in the problem to write, Furthermore, we can obtain an expression for by considering Putting these expressions for and into our original equation we have But this implies which is impossible. Hence, there can be no real satisfying this equation.

# Prove that the derivative of csch x is -csch x coth x

Prove the following formula for the derivative of the hyperbolic cosecant, Proof. We can compute the derivative directly, # Prove that the derivative of sech x is -sech x tanh x

Prove the following formula for the derivative of the hyperbolic secant, Proof. We can compute this directly, # Prove that the derivative of coth x is -csch2 x

Prove the following formula for the derivative of the hyperbolic cotangent, Proof. We know from the previous two exercises (here and here that Furthermore, we know from this exercise that So, we can compute the derivative # Prove that the derivative of tanh x is sech2x

Prove the following formula for the derivative of the hyperbolic tangent, Proof. We know from the previous two exercises (here and here that Furthermore, we know from this exercise that Thus, we can compute 