# Help figuring out this trigometric substitution

• ComFlu945
In summary, the conversation discusses the use of complex numbers and trigonometric functions to simplify the equation w=u(x+iy)*(x^2-y^2+k^2)^2+(2xy)^2)^-.25, with the variables N, M, R, and theta being used to represent different components of the equation. The final simplified form is shown to be w=u(z)*(cos(theta/2)-isin(theta/2)*(R^2)^-.25.
ComFlu945
From my notes I have

w=u(x+iy)*(x^2 - y^2 +k^2 + i(2xy))^-.5

We let N=x^2-y^2+k^2
M=2xy
R^2=(N^2+M^2)^2
theta=tan^-1(M/N)

using this, now

w=u(x+iy)*(cos(theta/2)-isin(theta/2))*(x^2 - y^2 +k^2 )^2 + (2xy)^2 )^-.25

I don't get that part. Btw, it simplifies to
w=u(x+iy)*(cos(theta/2)-isin(theta/2)*(R^2)^-.25

This has nothing to do with "Abstract and Linear Algebra" so I am moving it to "General Math".

I haven't thought about that particular problem, but note that
$$(x+iy)^2=x^2-y^2+2ixy$$

$$\frac{u(z)}{\sqrt{z^2+k^2}}$$

It seems you used $z^2+k^2=Re^{i\theta}[/tex] hence [itex]R=\abs{z^2+k^2}$ and $\theta=\arg(z^2+k^2)$

Last edited:
I figured it out. Let O=N+iM. Then let O=R*e^-i*theta

## 1. What is a trigonometric substitution?

A trigonometric substitution is a method used in calculus to solve integrals involving trigonometric functions. It involves replacing the variable in the integral with a trigonometric function and using trigonometric identities to simplify the integral.

## 2. When should I use a trigonometric substitution?

You should use a trigonometric substitution when you have an integral that contains a square root of a quadratic expression, or when you have an integral that contains the product of a trigonometric function and a power of a quadratic expression.

## 3. How do I choose which trigonometric substitution to use?

The choice of trigonometric substitution depends on the form of the integral. For integrals that contain a square root of a quadratic expression, you can use the substitution x = a sin θ or x = a tan θ. For integrals that contain the product of a trigonometric function and a power of a quadratic expression, you can use the substitution x = a sec θ or x = a cot θ.

## 4. What are the common trigonometric identities used in trigonometric substitution?

The common trigonometric identities used in trigonometric substitution include the Pythagorean identities, double angle identities, and half angle identities. These identities are used to simplify the integral and eliminate the trigonometric functions.

## 5. Are there any tips for solving integrals using trigonometric substitution?

Some tips for solving integrals using trigonometric substitution include carefully choosing the substitution, using the appropriate trigonometric identities, and being mindful of the limits of integration. It is also important to check your work and simplify the final answer using algebraic techniques.

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